[Pkg-privacy-commits] [golang-ed25519-dev] 02/11: Replace implementation with a port of ref10 code.

Ximin Luo infinity0 at moszumanska.debian.org
Sat Aug 22 12:55:25 UTC 2015


This is an automated email from the git hooks/post-receive script.

infinity0 pushed a commit to branch master
in repository golang-ed25519-dev.

commit 5613cf01b3df739fcda8f197edaf754fa9e627b1
Author: Adam Langley <agl at imperialviolet.org>
Date:   Mon Sep 2 11:53:39 2013 -0400

    Replace implementation with a port of ref10 code.
    
    This implementation is much faster and is now constant-time.
---
 const.go        | 1403 ++++++++++++++++++++++++++++++++++
 ed25519.go      | 2242 +++++++++++++++++++++++++++++++++++++++++++++++++++++++
 ed25519_ref.go  |  377 ----------
 ed25519_test.go |    3 +-
 4 files changed, 3646 insertions(+), 379 deletions(-)

diff --git a/const.go b/const.go
new file mode 100644
index 0000000..e534b55
--- /dev/null
+++ b/const.go
@@ -0,0 +1,1403 @@
+package ed25519
+
+var d = fieldElement{
+	-10913610, 13857413, -15372611, 6949391, 114729, -8787816, -6275908, -3247719, -18696448, -12055116,
+}
+
+var d2 = fieldElement{
+	-21827239, -5839606, -30745221, 13898782, 229458, 15978800, -12551817, -6495438, 29715968, 9444199,
+}
+
+var sqrtm1 = fieldElement{
+	-32595792, -7943725, 9377950, 3500415, 12389472, -272473, -25146209, -2005654, 326686, 11406482,
+}
+
+var bi = [8]preComputedGroupElement{
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+			fieldElement{-16657702, 2817643, -10286362, 14811298, 6024667, 13349505, -27315504, -10497842, -27672585, -11539858},
+			fieldElement{15941029, -9405932, -21367050, 8062055, 31876073, -238629, -15278393, -1444429, 15397331, -4130193},
+			fieldElement{8934485, -13485467, -23286397, -13423241, -32446090, 14047986, 31170398, -1441021, -27505566, 15087184},
+		},
+		{
+			fieldElement{-18357243, -2156491, 24524913, -16677868, 15520427, -6360776, -15502406, 11461896, 16788528, -5868942},
+			fieldElement{-1947386, 16013773, 21750665, 3714552, -17401782, -16055433, -3770287, -10323320, 31322514, -11615635},
+			fieldElement{21426655, -5650218, -13648287, -5347537, -28812189, -4920970, -18275391, -14621414, 13040862, -12112948},
+		},
+		{
+			fieldElement{11293895, 12478086, -27136401, 15083750, -29307421, 14748872, 14555558, -13417103, 1613711, 4896935},
+			fieldElement{-25894883, 15323294, -8489791, -8057900, 25967126, -13425460, 2825960, -4897045, -23971776, -11267415},
+			fieldElement{-15924766, -5229880, -17443532, 6410664, 3622847, 10243618, 20615400, 12405433, -23753030, -8436416},
+		},
+		{
+			fieldElement{-7091295, 12556208, -20191352, 9025187, -17072479, 4333801, 4378436, 2432030, 23097949, -566018},
+			fieldElement{4565804, -16025654, 20084412, -7842817, 1724999, 189254, 24767264, 10103221, -18512313, 2424778},
+			fieldElement{366633, -11976806, 8173090, -6890119, 30788634, 5745705, -7168678, 1344109, -3642553, 12412659},
+		},
+		{
+			fieldElement{-24001791, 7690286, 14929416, -168257, -32210835, -13412986, 24162697, -15326504, -3141501, 11179385},
+			fieldElement{18289522, -14724954, 8056945, 16430056, -21729724, 7842514, -6001441, -1486897, -18684645, -11443503},
+			fieldElement{476239, 6601091, -6152790, -9723375, 17503545, -4863900, 27672959, 13403813, 11052904, 5219329},
+		},
+	},
+	{
+		{
+			fieldElement{20678546, -8375738, -32671898, 8849123, -5009758, 14574752, 31186971, -3973730, 9014762, -8579056},
+			fieldElement{-13644050, -10350239, -15962508, 5075808, -1514661, -11534600, -33102500, 9160280, 8473550, -3256838},
+			fieldElement{24900749, 14435722, 17209120, -15292541, -22592275, 9878983, -7689309, -16335821, -24568481, 11788948},
+		},
+		{
+			fieldElement{-3118155, -11395194, -13802089, 14797441, 9652448, -6845904, -20037437, 10410733, -24568470, -1458691},
+			fieldElement{-15659161, 16736706, -22467150, 10215878, -9097177, 7563911, 11871841, -12505194, -18513325, 8464118},
+			fieldElement{-23400612, 8348507, -14585951, -861714, -3950205, -6373419, 14325289, 8628612, 33313881, -8370517},
+		},
+		{
+			fieldElement{-20186973, -4967935, 22367356, 5271547, -1097117, -4788838, -24805667, -10236854, -8940735, -5818269},
+			fieldElement{-6948785, -1795212, -32625683, -16021179, 32635414, -7374245, 15989197, -12838188, 28358192, -4253904},
+			fieldElement{-23561781, -2799059, -32351682, -1661963, -9147719, 10429267, -16637684, 4072016, -5351664, 5596589},
+		},
+		{
+			fieldElement{-28236598, -3390048, 12312896, 6213178, 3117142, 16078565, 29266239, 2557221, 1768301, 15373193},
+			fieldElement{-7243358, -3246960, -4593467, -7553353, -127927, -912245, -1090902, -4504991, -24660491, 3442910},
+			fieldElement{-30210571, 5124043, 14181784, 8197961, 18964734, -11939093, 22597931, 7176455, -18585478, 13365930},
+		},
+		{
+			fieldElement{-7877390, -1499958, 8324673, 4690079, 6261860, 890446, 24538107, -8570186, -9689599, -3031667},
+			fieldElement{25008904, -10771599, -4305031, -9638010, 16265036, 15721635, 683793, -11823784, 15723479, -15163481},
+			fieldElement{-9660625, 12374379, -27006999, -7026148, -7724114, -12314514, 11879682, 5400171, 519526, -1235876},
+		},
+		{
+			fieldElement{22258397, -16332233, -7869817, 14613016, -22520255, -2950923, -20353881, 7315967, 16648397, 7605640},
+			fieldElement{-8081308, -8464597, -8223311, 9719710, 19259459, -15348212, 23994942, -5281555, -9468848, 4763278},
+			fieldElement{-21699244, 9220969, -15730624, 1084137, -25476107, -2852390, 31088447, -7764523, -11356529, 728112},
+		},
+		{
+			fieldElement{26047220, -11751471, -6900323, -16521798, 24092068, 9158119, -4273545, -12555558, -29365436, -5498272},
+			fieldElement{17510331, -322857, 5854289, 8403524, 17133918, -3112612, -28111007, 12327945, 10750447, 10014012},
+			fieldElement{-10312768, 3936952, 9156313, -8897683, 16498692, -994647, -27481051, -666732, 3424691, 7540221},
+		},
+		{
+			fieldElement{30322361, -6964110, 11361005, -4143317, 7433304, 4989748, -7071422, -16317219, -9244265, 15258046},
+			fieldElement{13054562, -2779497, 19155474, 469045, -12482797, 4566042, 5631406, 2711395, 1062915, -5136345},
+			fieldElement{-19240248, -11254599, -29509029, -7499965, -5835763, 13005411, -6066489, 12194497, 32960380, 1459310},
+		},
+	},
+	{
+		{
+			fieldElement{19852034, 7027924, 23669353, 10020366, 8586503, -6657907, 394197, -6101885, 18638003, -11174937},
+			fieldElement{31395534, 15098109, 26581030, 8030562, -16527914, -5007134, 9012486, -7584354, -6643087, -5442636},
+			fieldElement{-9192165, -2347377, -1997099, 4529534, 25766844, 607986, -13222, 9677543, -32294889, -6456008},
+		},
+		{
+			fieldElement{-2444496, -149937, 29348902, 8186665, 1873760, 12489863, -30934579, -7839692, -7852844, -8138429},
+			fieldElement{-15236356, -15433509, 7766470, 746860, 26346930, -10221762, -27333451, 10754588, -9431476, 5203576},
+			fieldElement{31834314, 14135496, -770007, 5159118, 20917671, -16768096, -7467973, -7337524, 31809243, 7347066},
+		},
+		{
+			fieldElement{-9606723, -11874240, 20414459, 13033986, 13716524, -11691881, 19797970, -12211255, 15192876, -2087490},
+			fieldElement{-12663563, -2181719, 1168162, -3804809, 26747877, -14138091, 10609330, 12694420, 33473243, -13382104},
+			fieldElement{33184999, 11180355, 15832085, -11385430, -1633671, 225884, 15089336, -11023903, -6135662, 14480053},
+		},
+		{
+			fieldElement{31308717, -5619998, 31030840, -1897099, 15674547, -6582883, 5496208, 13685227, 27595050, 8737275},
+			fieldElement{-20318852, -15150239, 10933843, -16178022, 8335352, -7546022, -31008351, -12610604, 26498114, 66511},
+			fieldElement{22644454, -8761729, -16671776, 4884562, -3105614, -13559366, 30540766, -4286747, -13327787, -7515095},
+		},
+		{
+			fieldElement{-28017847, 9834845, 18617207, -2681312, -3401956, -13307506, 8205540, 13585437, -17127465, 15115439},
+			fieldElement{23711543, -672915, 31206561, -8362711, 6164647, -9709987, -33535882, -1426096, 8236921, 16492939},
+			fieldElement{-23910559, -13515526, -26299483, -4503841, 25005590, -7687270, 19574902, 10071562, 6708380, -6222424},
+		},
+		{
+			fieldElement{2101391, -4930054, 19702731, 2367575, -15427167, 1047675, 5301017, 9328700, 29955601, -11678310},
+			fieldElement{3096359, 9271816, -21620864, -15521844, -14847996, -7592937, -25892142, -12635595, -9917575, 6216608},
+			fieldElement{-32615849, 338663, -25195611, 2510422, -29213566, -13820213, 24822830, -6146567, -26767480, 7525079},
+		},
+		{
+			fieldElement{-23066649, -13985623, 16133487, -7896178, -3389565, 778788, -910336, -2782495, -19386633, 11994101},
+			fieldElement{21691500, -13624626, -641331, -14367021, 3285881, -3483596, -25064666, 9718258, -7477437, 13381418},
+			fieldElement{18445390, -4202236, 14979846, 11622458, -1727110, -3582980, 23111648, -6375247, 28535282, 15779576},
+		},
+		{
+			fieldElement{30098053, 3089662, -9234387, 16662135, -21306940, 11308411, -14068454, 12021730, 9955285, -16303356},
+			fieldElement{9734894, -14576830, -7473633, -9138735, 2060392, 11313496, -18426029, 9924399, 20194861, 13380996},
+			fieldElement{-26378102, -7965207, -22167821, 15789297, -18055342, -6168792, -1984914, 15707771, 26342023, 10146099},
+		},
+	},
+	{
+		{
+			fieldElement{-26016874, -219943, 21339191, -41388, 19745256, -2878700, -29637280, 2227040, 21612326, -545728},
+			fieldElement{-13077387, 1184228, 23562814, -5970442, -20351244, -6348714, 25764461, 12243797, -20856566, 11649658},
+			fieldElement{-10031494, 11262626, 27384172, 2271902, 26947504, -15997771, 39944, 6114064, 33514190, 2333242},
+		},
+		{
+			fieldElement{-21433588, -12421821, 8119782, 7219913, -21830522, -9016134, -6679750, -12670638, 24350578, -13450001},
+			fieldElement{-4116307, -11271533, -23886186, 4843615, -30088339, 690623, -31536088, -10406836, 8317860, 12352766},
+			fieldElement{18200138, -14475911, -33087759, -2696619, -23702521, -9102511, -23552096, -2287550, 20712163, 6719373},
+		},
+		{
+			fieldElement{26656208, 6075253, -7858556, 1886072, -28344043, 4262326, 11117530, -3763210, 26224235, -3297458},
+			fieldElement{-17168938, -14854097, -3395676, -16369877, -19954045, 14050420, 21728352, 9493610, 18620611, -16428628},
+			fieldElement{-13323321, 13325349, 11432106, 5964811, 18609221, 6062965, -5269471, -9725556, -30701573, -16479657},
+		},
+		{
+			fieldElement{-23860538, -11233159, 26961357, 1640861, -32413112, -16737940, 12248509, -5240639, 13735342, 1934062},
+			fieldElement{25089769, 6742589, 17081145, -13406266, 21909293, -16067981, -15136294, -3765346, -21277997, 5473616},
+			fieldElement{31883677, -7961101, 1083432, -11572403, 22828471, 13290673, -7125085, 12469656, 29111212, -5451014},
+		},
+		{
+			fieldElement{24244947, -15050407, -26262976, 2791540, -14997599, 16666678, 24367466, 6388839, -10295587, 452383},
+			fieldElement{-25640782, -3417841, 5217916, 16224624, 19987036, -4082269, -24236251, -5915248, 15766062, 8407814},
+			fieldElement{-20406999, 13990231, 15495425, 16395525, 5377168, 15166495, -8917023, -4388953, -8067909, 2276718},
+		},
+		{
+			fieldElement{30157918, 12924066, -17712050, 9245753, 19895028, 3368142, -23827587, 5096219, 22740376, -7303417},
+			fieldElement{2041139, -14256350, 7783687, 13876377, -25946985, -13352459, 24051124, 13742383, -15637599, 13295222},
+			fieldElement{33338237, -8505733, 12532113, 7977527, 9106186, -1715251, -17720195, -4612972, -4451357, -14669444},
+		},
+		{
+			fieldElement{-20045281, 5454097, -14346548, 6447146, 28862071, 1883651, -2469266, -4141880, 7770569, 9620597},
+			fieldElement{23208068, 7979712, 33071466, 8149229, 1758231, -10834995, 30945528, -1694323, -33502340, -14767970},
+			fieldElement{1439958, -16270480, -1079989, -793782, 4625402, 10647766, -5043801, 1220118, 30494170, -11440799},
+		},
+		{
+			fieldElement{-5037580, -13028295, -2970559, -3061767, 15640974, -6701666, -26739026, 926050, -1684339, -13333647},
+			fieldElement{13908495, -3549272, 30919928, -6273825, -21521863, 7989039, 9021034, 9078865, 3353509, 4033511},
+			fieldElement{-29663431, -15113610, 32259991, -344482, 24295849, -12912123, 23161163, 8839127, 27485041, 7356032},
+		},
+	},
+	{
+		{
+			fieldElement{9661027, 705443, 11980065, -5370154, -1628543, 14661173, -6346142, 2625015, 28431036, -16771834},
+			fieldElement{-23839233, -8311415, -25945511, 7480958, -17681669, -8354183, -22545972, 14150565, 15970762, 4099461},
+			fieldElement{29262576, 16756590, 26350592, -8793563, 8529671, -11208050, 13617293, -9937143, 11465739, 8317062},
+		},
+		{
+			fieldElement{-25493081, -6962928, 32500200, -9419051, -23038724, -2302222, 14898637, 3848455, 20969334, -5157516},
+			fieldElement{-20384450, -14347713, -18336405, 13884722, -33039454, 2842114, -21610826, -3649888, 11177095, 14989547},
+			fieldElement{-24496721, -11716016, 16959896, 2278463, 12066309, 10137771, 13515641, 2581286, -28487508, 9930240},
+		},
+		{
+			fieldElement{-17751622, -2097826, 16544300, -13009300, -15914807, -14949081, 18345767, -13403753, 16291481, -5314038},
+			fieldElement{-33229194, 2553288, 32678213, 9875984, 8534129, 6889387, -9676774, 6957617, 4368891, 9788741},
+			fieldElement{16660756, 7281060, -10830758, 12911820, 20108584, -8101676, -21722536, -8613148, 16250552, -11111103},
+		},
+		{
+			fieldElement{-19765507, 2390526, -16551031, 14161980, 1905286, 6414907, 4689584, 10604807, -30190403, 4782747},
+			fieldElement{-1354539, 14736941, -7367442, -13292886, 7710542, -14155590, -9981571, 4383045, 22546403, 437323},
+			fieldElement{31665577, -12180464, -16186830, 1491339, -18368625, 3294682, 27343084, 2786261, -30633590, -14097016},
+		},
+		{
+			fieldElement{-14467279, -683715, -33374107, 7448552, 19294360, 14334329, -19690631, 2355319, -19284671, -6114373},
+			fieldElement{15121312, -15796162, 6377020, -6031361, -10798111, -12957845, 18952177, 15496498, -29380133, 11754228},
+			fieldElement{-2637277, -13483075, 8488727, -14303896, 12728761, -1622493, 7141596, 11724556, 22761615, -10134141},
+		},
+		{
+			fieldElement{16918416, 11729663, -18083579, 3022987, -31015732, -13339659, -28741185, -12227393, 32851222, 11717399},
+			fieldElement{11166634, 7338049, -6722523, 4531520, -29468672, -7302055, 31474879, 3483633, -1193175, -4030831},
+			fieldElement{-185635, 9921305, 31456609, -13536438, -12013818, 13348923, 33142652, 6546660, -19985279, -3948376},
+		},
+		{
+			fieldElement{-32460596, 11266712, -11197107, -7899103, 31703694, 3855903, -8537131, -12833048, -30772034, -15486313},
+			fieldElement{-18006477, 12709068, 3991746, -6479188, -21491523, -10550425, -31135347, -16049879, 10928917, 3011958},
+			fieldElement{-6957757, -15594337, 31696059, 334240, 29576716, 14796075, -30831056, -12805180, 18008031, 10258577},
+		},
+		{
+			fieldElement{-22448644, 15655569, 7018479, -4410003, -30314266, -1201591, -1853465, 1367120, 25127874, 6671743},
+			fieldElement{29701166, -14373934, -10878120, 9279288, -17568, 13127210, 21382910, 11042292, 25838796, 4642684},
+			fieldElement{-20430234, 14955537, -24126347, 8124619, -5369288, -5990470, 30468147, -13900640, 18423289, 4177476},
+		},
+	},
+}
diff --git a/ed25519.go b/ed25519.go
new file mode 100644
index 0000000..9013eb3
--- /dev/null
+++ b/ed25519.go
@@ -0,0 +1,2242 @@
+// Copyright 2013 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// Package ed25519 implements the Ed25519 signature algorithm. See
+// http://ed25519.cr.yp.to/.
+package ed25519
+
+// This code is a port of the public domain, "ref10" implementation of ed25519
+// from SUPERCOP.
+
+import (
+	"bytes"
+	"crypto/sha512"
+	"io"
+)
+
+const (
+	PublicKeySize  = 32
+	PrivateKeySize = 64
+	SignatureSize  = 64
+)
+
+// fieldElement represents an element of the field GF(2^255 - 19).  An element
+// t, entries t[0]...t[9], represents the integer t[0]+2^26 t[1]+2^51 t[2]+2^77
+// t[3]+2^102 t[4]+...+2^230 t[9].  Bounds on each t[i] vary depending on
+// context.
+type fieldElement [10]int32
+
+func feZero(fe *fieldElement) {
+	for i := range fe {
+		fe[i] = 0
+	}
+}
+
+func feOne(fe *fieldElement) {
+	feZero(fe)
+	fe[0] = 1
+}
+
+func feAdd(dst, a, b *fieldElement) {
+	for i := range dst {
+		dst[i] = a[i] + b[i]
+	}
+}
+
+func feSub(dst, a, b *fieldElement) {
+	for i := range dst {
+		dst[i] = a[i] - b[i]
+	}
+}
+
+func feCopy(dst, src *fieldElement) {
+	for i := range dst {
+		dst[i] = src[i]
+	}
+}
+
+// Replace (f,g) with (g,g) if b == 1;
+// replace (f,g) with (f,g) if b == 0.
+//
+// Preconditions: b in {0,1}.
+func feCMove(f, g *fieldElement, b int32) {
+	var x fieldElement
+	b = -b
+	for i := range x {
+		x[i] = b & (f[i] ^ g[i])
+	}
+
+	for i := range f {
+		f[i] ^= x[i]
+	}
+}
+
+func load3(in []byte) int64 {
+	var r int64
+	r = int64(in[0])
+	r |= int64(in[1]) << 8
+	r |= int64(in[2]) << 16
+	return r
+}
+
+func load4(in []byte) int64 {
+	var r int64
+	r = int64(in[0])
+	r |= int64(in[1]) << 8
+	r |= int64(in[2]) << 16
+	r |= int64(in[3]) << 24
+	return r
+}
+
+func feFromBytes(dst *fieldElement, src *[32]byte) {
+	h0 := load4(src[:])
+	h1 := load3(src[4:]) << 6
+	h2 := load3(src[7:]) << 5
+	h3 := load3(src[10:]) << 3
+	h4 := load3(src[13:]) << 2
+	h5 := load4(src[16:])
+	h6 := load3(src[20:]) << 7
+	h7 := load3(src[23:]) << 5
+	h8 := load3(src[26:]) << 4
+	h9 := (load3(src[29:]) & 8388607) << 2
+
+	var carry [10]int64
+	carry[9] = (h9 + 1<<24) >> 25
+	h0 += carry[9] * 19
+	h9 -= carry[9] << 25
+	carry[1] = (h1 + 1<<24) >> 25
+	h2 += carry[1]
+	h1 -= carry[1] << 25
+	carry[3] = (h3 + 1<<24) >> 25
+	h4 += carry[3]
+	h3 -= carry[3] << 25
+	carry[5] = (h5 + 1<<24) >> 25
+	h6 += carry[5]
+	h5 -= carry[5] << 25
+	carry[7] = (h7 + 1<<24) >> 25
+	h8 += carry[7]
+	h7 -= carry[7] << 25
+
+	carry[0] = (h0 + 1<<25) >> 26
+	h1 += carry[0]
+	h0 -= carry[0] << 26
+	carry[2] = (h2 + 1<<25) >> 26
+	h3 += carry[2]
+	h2 -= carry[2] << 26
+	carry[4] = (h4 + 1<<25) >> 26
+	h5 += carry[4]
+	h4 -= carry[4] << 26
+	carry[6] = (h6 + 1<<25) >> 26
+	h7 += carry[6]
+	h6 -= carry[6] << 26
+	carry[8] = (h8 + 1<<25) >> 26
+	h9 += carry[8]
+	h8 -= carry[8] << 26
+
+	dst[0] = int32(h0)
+	dst[1] = int32(h1)
+	dst[2] = int32(h2)
+	dst[3] = int32(h3)
+	dst[4] = int32(h4)
+	dst[5] = int32(h5)
+	dst[6] = int32(h6)
+	dst[7] = int32(h7)
+	dst[8] = int32(h8)
+	dst[9] = int32(h9)
+}
+
+// feToBytes marshals h to s.
+// Preconditions:
+//   |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+//
+// Write p=2^255-19; q=floor(h/p).
+// Basic claim: q = floor(2^(-255)(h + 19 2^(-25)h9 + 2^(-1))).
+//
+// Proof:
+//   Have |h|<=p so |q|<=1 so |19^2 2^(-255) q|<1/4.
+//   Also have |h-2^230 h9|<2^230 so |19 2^(-255)(h-2^230 h9)|<1/4.
+//
+//   Write y=2^(-1)-19^2 2^(-255)q-19 2^(-255)(h-2^230 h9).
+//   Then 0<y<1.
+//
+//   Write r=h-pq.
+//   Have 0<=r<=p-1=2^255-20.
+//   Thus 0<=r+19(2^-255)r<r+19(2^-255)2^255<=2^255-1.
+//
+//   Write x=r+19(2^-255)r+y.
+//   Then 0<x<2^255 so floor(2^(-255)x) = 0 so floor(q+2^(-255)x) = q.
+//
+//   Have q+2^(-255)x = 2^(-255)(h + 19 2^(-25) h9 + 2^(-1))
+//   so floor(2^(-255)(h + 19 2^(-25) h9 + 2^(-1))) = q.
+func feToBytes(s *[32]byte, h *fieldElement) {
+	var carry [10]int32
+
+	q := (19*h[9] + (1 << 24)) >> 25
+	q = (h[0] + q) >> 26
+	q = (h[1] + q) >> 25
+	q = (h[2] + q) >> 26
+	q = (h[3] + q) >> 25
+	q = (h[4] + q) >> 26
+	q = (h[5] + q) >> 25
+	q = (h[6] + q) >> 26
+	q = (h[7] + q) >> 25
+	q = (h[8] + q) >> 26
+	q = (h[9] + q) >> 25
+
+	// Goal: Output h-(2^255-19)q, which is between 0 and 2^255-20.
+	h[0] += 19 * q
+	// Goal: Output h-2^255 q, which is between 0 and 2^255-20.
+
+	carry[0] = h[0] >> 26
+	h[1] += carry[0]
+	h[0] -= carry[0] << 26
+	carry[1] = h[1] >> 25
+	h[2] += carry[1]
+	h[1] -= carry[1] << 25
+	carry[2] = h[2] >> 26
+	h[3] += carry[2]
+	h[2] -= carry[2] << 26
+	carry[3] = h[3] >> 25
+	h[4] += carry[3]
+	h[3] -= carry[3] << 25
+	carry[4] = h[4] >> 26
+	h[5] += carry[4]
+	h[4] -= carry[4] << 26
+	carry[5] = h[5] >> 25
+	h[6] += carry[5]
+	h[5] -= carry[5] << 25
+	carry[6] = h[6] >> 26
+	h[7] += carry[6]
+	h[6] -= carry[6] << 26
+	carry[7] = h[7] >> 25
+	h[8] += carry[7]
+	h[7] -= carry[7] << 25
+	carry[8] = h[8] >> 26
+	h[9] += carry[8]
+	h[8] -= carry[8] << 26
+	carry[9] = h[9] >> 25
+	h[9] -= carry[9] << 25
+	// h10 = carry9
+
+	// Goal: Output h[0]+...+2^255 h10-2^255 q, which is between 0 and 2^255-20.
+	// Have h[0]+...+2^230 h[9] between 0 and 2^255-1;
+	// evidently 2^255 h10-2^255 q = 0.
+	// Goal: Output h[0]+...+2^230 h[9].
+
+	s[0] = byte(h[0] >> 0)
+	s[1] = byte(h[0] >> 8)
+	s[2] = byte(h[0] >> 16)
+	s[3] = byte((h[0] >> 24) | (h[1] << 2))
+	s[4] = byte(h[1] >> 6)
+	s[5] = byte(h[1] >> 14)
+	s[6] = byte((h[1] >> 22) | (h[2] << 3))
+	s[7] = byte(h[2] >> 5)
+	s[8] = byte(h[2] >> 13)
+	s[9] = byte((h[2] >> 21) | (h[3] << 5))
+	s[10] = byte(h[3] >> 3)
+	s[11] = byte(h[3] >> 11)
+	s[12] = byte((h[3] >> 19) | (h[4] << 6))
+	s[13] = byte(h[4] >> 2)
+	s[14] = byte(h[4] >> 10)
+	s[15] = byte(h[4] >> 18)
+	s[16] = byte(h[5] >> 0)
+	s[17] = byte(h[5] >> 8)
+	s[18] = byte(h[5] >> 16)
+	s[19] = byte((h[5] >> 24) | (h[6] << 1))
+	s[20] = byte(h[6] >> 7)
+	s[21] = byte(h[6] >> 15)
+	s[22] = byte((h[6] >> 23) | (h[7] << 3))
+	s[23] = byte(h[7] >> 5)
+	s[24] = byte(h[7] >> 13)
+	s[25] = byte((h[7] >> 21) | (h[8] << 4))
+	s[26] = byte(h[8] >> 4)
+	s[27] = byte(h[8] >> 12)
+	s[28] = byte((h[8] >> 20) | (h[9] << 6))
+	s[29] = byte(h[9] >> 2)
+	s[30] = byte(h[9] >> 10)
+	s[31] = byte(h[9] >> 18)
+}
+
+func feIsNegative(f *fieldElement) byte {
+	var s [32]byte
+	feToBytes(&s, f)
+	return s[0] & 1
+}
+
+func feIsNonZero(f *fieldElement) bool {
+	var s [32]byte
+	feToBytes(&s, f)
+	var x uint8
+	for _, b := range s {
+		x |= b
+	}
+	return x != 0
+}
+
+// feNeg sets h = -f
+//
+// Preconditions:
+//    |f| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+//
+// Postconditions:
+//    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+func feNeg(h, f *fieldElement) {
+	for i := range h {
+		h[i] = -f[i]
+	}
+}
+
+// feMul calculates h = f * g
+// Can overlap h with f or g.
+//
+// Preconditions:
+//    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
+//    |g| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
+//
+// Postconditions:
+//    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+//
+// Notes on implementation strategy:
+//
+// Using schoolbook multiplication.
+// Karatsuba would save a little in some cost models.
+//
+// Most multiplications by 2 and 19 are 32-bit precomputations;
+// cheaper than 64-bit postcomputations.
+//
+// There is one remaining multiplication by 19 in the carry chain;
+// one *19 precomputation can be merged into this,
+// but the resulting data flow is considerably less clean.
+//
+// There are 12 carries below.
+// 10 of them are 2-way parallelizable and vectorizable.
+// Can get away with 11 carries, but then data flow is much deeper.
+//
+// With tighter constraints on inputs can squeeze carries into int32.
+func feMul(h, f, g *fieldElement) {
+	f0 := f[0]
+	f1 := f[1]
+	f2 := f[2]
+	f3 := f[3]
+	f4 := f[4]
+	f5 := f[5]
+	f6 := f[6]
+	f7 := f[7]
+	f8 := f[8]
+	f9 := f[9]
+	g0 := g[0]
+	g1 := g[1]
+	g2 := g[2]
+	g3 := g[3]
+	g4 := g[4]
+	g5 := g[5]
+	g6 := g[6]
+	g7 := g[7]
+	g8 := g[8]
+	g9 := g[9]
+	g1_19 := 19 * g1 /* 1.4*2^29 */
+	g2_19 := 19 * g2 /* 1.4*2^30; still ok */
+	g3_19 := 19 * g3
+	g4_19 := 19 * g4
+	g5_19 := 19 * g5
+	g6_19 := 19 * g6
+	g7_19 := 19 * g7
+	g8_19 := 19 * g8
+	g9_19 := 19 * g9
+	f1_2 := 2 * f1
+	f3_2 := 2 * f3
+	f5_2 := 2 * f5
+	f7_2 := 2 * f7
+	f9_2 := 2 * f9
+	f0g0 := int64(f0) * int64(g0)
+	f0g1 := int64(f0) * int64(g1)
+	f0g2 := int64(f0) * int64(g2)
+	f0g3 := int64(f0) * int64(g3)
+	f0g4 := int64(f0) * int64(g4)
+	f0g5 := int64(f0) * int64(g5)
+	f0g6 := int64(f0) * int64(g6)
+	f0g7 := int64(f0) * int64(g7)
+	f0g8 := int64(f0) * int64(g8)
+	f0g9 := int64(f0) * int64(g9)
+	f1g0 := int64(f1) * int64(g0)
+	f1g1_2 := int64(f1_2) * int64(g1)
+	f1g2 := int64(f1) * int64(g2)
+	f1g3_2 := int64(f1_2) * int64(g3)
+	f1g4 := int64(f1) * int64(g4)
+	f1g5_2 := int64(f1_2) * int64(g5)
+	f1g6 := int64(f1) * int64(g6)
+	f1g7_2 := int64(f1_2) * int64(g7)
+	f1g8 := int64(f1) * int64(g8)
+	f1g9_38 := int64(f1_2) * int64(g9_19)
+	f2g0 := int64(f2) * int64(g0)
+	f2g1 := int64(f2) * int64(g1)
+	f2g2 := int64(f2) * int64(g2)
+	f2g3 := int64(f2) * int64(g3)
+	f2g4 := int64(f2) * int64(g4)
+	f2g5 := int64(f2) * int64(g5)
+	f2g6 := int64(f2) * int64(g6)
+	f2g7 := int64(f2) * int64(g7)
+	f2g8_19 := int64(f2) * int64(g8_19)
+	f2g9_19 := int64(f2) * int64(g9_19)
+	f3g0 := int64(f3) * int64(g0)
+	f3g1_2 := int64(f3_2) * int64(g1)
+	f3g2 := int64(f3) * int64(g2)
+	f3g3_2 := int64(f3_2) * int64(g3)
+	f3g4 := int64(f3) * int64(g4)
+	f3g5_2 := int64(f3_2) * int64(g5)
+	f3g6 := int64(f3) * int64(g6)
+	f3g7_38 := int64(f3_2) * int64(g7_19)
+	f3g8_19 := int64(f3) * int64(g8_19)
+	f3g9_38 := int64(f3_2) * int64(g9_19)
+	f4g0 := int64(f4) * int64(g0)
+	f4g1 := int64(f4) * int64(g1)
+	f4g2 := int64(f4) * int64(g2)
+	f4g3 := int64(f4) * int64(g3)
+	f4g4 := int64(f4) * int64(g4)
+	f4g5 := int64(f4) * int64(g5)
+	f4g6_19 := int64(f4) * int64(g6_19)
+	f4g7_19 := int64(f4) * int64(g7_19)
+	f4g8_19 := int64(f4) * int64(g8_19)
+	f4g9_19 := int64(f4) * int64(g9_19)
+	f5g0 := int64(f5) * int64(g0)
+	f5g1_2 := int64(f5_2) * int64(g1)
+	f5g2 := int64(f5) * int64(g2)
+	f5g3_2 := int64(f5_2) * int64(g3)
+	f5g4 := int64(f5) * int64(g4)
+	f5g5_38 := int64(f5_2) * int64(g5_19)
+	f5g6_19 := int64(f5) * int64(g6_19)
+	f5g7_38 := int64(f5_2) * int64(g7_19)
+	f5g8_19 := int64(f5) * int64(g8_19)
+	f5g9_38 := int64(f5_2) * int64(g9_19)
+	f6g0 := int64(f6) * int64(g0)
+	f6g1 := int64(f6) * int64(g1)
+	f6g2 := int64(f6) * int64(g2)
+	f6g3 := int64(f6) * int64(g3)
+	f6g4_19 := int64(f6) * int64(g4_19)
+	f6g5_19 := int64(f6) * int64(g5_19)
+	f6g6_19 := int64(f6) * int64(g6_19)
+	f6g7_19 := int64(f6) * int64(g7_19)
+	f6g8_19 := int64(f6) * int64(g8_19)
+	f6g9_19 := int64(f6) * int64(g9_19)
+	f7g0 := int64(f7) * int64(g0)
+	f7g1_2 := int64(f7_2) * int64(g1)
+	f7g2 := int64(f7) * int64(g2)
+	f7g3_38 := int64(f7_2) * int64(g3_19)
+	f7g4_19 := int64(f7) * int64(g4_19)
+	f7g5_38 := int64(f7_2) * int64(g5_19)
+	f7g6_19 := int64(f7) * int64(g6_19)
+	f7g7_38 := int64(f7_2) * int64(g7_19)
+	f7g8_19 := int64(f7) * int64(g8_19)
+	f7g9_38 := int64(f7_2) * int64(g9_19)
+	f8g0 := int64(f8) * int64(g0)
+	f8g1 := int64(f8) * int64(g1)
+	f8g2_19 := int64(f8) * int64(g2_19)
+	f8g3_19 := int64(f8) * int64(g3_19)
+	f8g4_19 := int64(f8) * int64(g4_19)
+	f8g5_19 := int64(f8) * int64(g5_19)
+	f8g6_19 := int64(f8) * int64(g6_19)
+	f8g7_19 := int64(f8) * int64(g7_19)
+	f8g8_19 := int64(f8) * int64(g8_19)
+	f8g9_19 := int64(f8) * int64(g9_19)
+	f9g0 := int64(f9) * int64(g0)
+	f9g1_38 := int64(f9_2) * int64(g1_19)
+	f9g2_19 := int64(f9) * int64(g2_19)
+	f9g3_38 := int64(f9_2) * int64(g3_19)
+	f9g4_19 := int64(f9) * int64(g4_19)
+	f9g5_38 := int64(f9_2) * int64(g5_19)
+	f9g6_19 := int64(f9) * int64(g6_19)
+	f9g7_38 := int64(f9_2) * int64(g7_19)
+	f9g8_19 := int64(f9) * int64(g8_19)
+	f9g9_38 := int64(f9_2) * int64(g9_19)
+	h0 := f0g0 + f1g9_38 + f2g8_19 + f3g7_38 + f4g6_19 + f5g5_38 + f6g4_19 + f7g3_38 + f8g2_19 + f9g1_38
+	h1 := f0g1 + f1g0 + f2g9_19 + f3g8_19 + f4g7_19 + f5g6_19 + f6g5_19 + f7g4_19 + f8g3_19 + f9g2_19
+	h2 := f0g2 + f1g1_2 + f2g0 + f3g9_38 + f4g8_19 + f5g7_38 + f6g6_19 + f7g5_38 + f8g4_19 + f9g3_38
+	h3 := f0g3 + f1g2 + f2g1 + f3g0 + f4g9_19 + f5g8_19 + f6g7_19 + f7g6_19 + f8g5_19 + f9g4_19
+	h4 := f0g4 + f1g3_2 + f2g2 + f3g1_2 + f4g0 + f5g9_38 + f6g8_19 + f7g7_38 + f8g6_19 + f9g5_38
+	h5 := f0g5 + f1g4 + f2g3 + f3g2 + f4g1 + f5g0 + f6g9_19 + f7g8_19 + f8g7_19 + f9g6_19
+	h6 := f0g6 + f1g5_2 + f2g4 + f3g3_2 + f4g2 + f5g1_2 + f6g0 + f7g9_38 + f8g8_19 + f9g7_38
+	h7 := f0g7 + f1g6 + f2g5 + f3g4 + f4g3 + f5g2 + f6g1 + f7g0 + f8g9_19 + f9g8_19
+	h8 := f0g8 + f1g7_2 + f2g6 + f3g5_2 + f4g4 + f5g3_2 + f6g2 + f7g1_2 + f8g0 + f9g9_38
+	h9 := f0g9 + f1g8 + f2g7 + f3g6 + f4g5 + f5g4 + f6g3 + f7g2 + f8g1 + f9g0
+	var carry [10]int64
+
+	/*
+	  |h0| <= (1.1*1.1*2^52*(1+19+19+19+19)+1.1*1.1*2^50*(38+38+38+38+38))
+	    i.e. |h0| <= 1.2*2^59; narrower ranges for h2, h4, h6, h8
+	  |h1| <= (1.1*1.1*2^51*(1+1+19+19+19+19+19+19+19+19))
+	    i.e. |h1| <= 1.5*2^58; narrower ranges for h3, h5, h7, h9
+	*/
+
+	carry[0] = (h0 + (1 << 25)) >> 26
+	h1 += carry[0]
+	h0 -= carry[0] << 26
+	carry[4] = (h4 + (1 << 25)) >> 26
+	h5 += carry[4]
+	h4 -= carry[4] << 26
+	/* |h0| <= 2^25 */
+	/* |h4| <= 2^25 */
+	/* |h1| <= 1.51*2^58 */
+	/* |h5| <= 1.51*2^58 */
+
+	carry[1] = (h1 + (1 << 24)) >> 25
+	h2 += carry[1]
+	h1 -= carry[1] << 25
+	carry[5] = (h5 + (1 << 24)) >> 25
+	h6 += carry[5]
+	h5 -= carry[5] << 25
+	/* |h1| <= 2^24; from now on fits into int32 */
+	/* |h5| <= 2^24; from now on fits into int32 */
+	/* |h2| <= 1.21*2^59 */
+	/* |h6| <= 1.21*2^59 */
+
+	carry[2] = (h2 + (1 << 25)) >> 26
+	h3 += carry[2]
+	h2 -= carry[2] << 26
+	carry[6] = (h6 + (1 << 25)) >> 26
+	h7 += carry[6]
+	h6 -= carry[6] << 26
+	/* |h2| <= 2^25; from now on fits into int32 unchanged */
+	/* |h6| <= 2^25; from now on fits into int32 unchanged */
+	/* |h3| <= 1.51*2^58 */
+	/* |h7| <= 1.51*2^58 */
+
+	carry[3] = (h3 + (1 << 24)) >> 25
+	h4 += carry[3]
+	h3 -= carry[3] << 25
+	carry[7] = (h7 + (1 << 24)) >> 25
+	h8 += carry[7]
+	h7 -= carry[7] << 25
+	/* |h3| <= 2^24; from now on fits into int32 unchanged */
+	/* |h7| <= 2^24; from now on fits into int32 unchanged */
+	/* |h4| <= 1.52*2^33 */
+	/* |h8| <= 1.52*2^33 */
+
+	carry[4] = (h4 + (1 << 25)) >> 26
+	h5 += carry[4]
+	h4 -= carry[4] << 26
+	carry[8] = (h8 + (1 << 25)) >> 26
+	h9 += carry[8]
+	h8 -= carry[8] << 26
+	/* |h4| <= 2^25; from now on fits into int32 unchanged */
+	/* |h8| <= 2^25; from now on fits into int32 unchanged */
+	/* |h5| <= 1.01*2^24 */
+	/* |h9| <= 1.51*2^58 */
+
+	carry[9] = (h9 + (1 << 24)) >> 25
+	h0 += carry[9] * 19
+	h9 -= carry[9] << 25
+	/* |h9| <= 2^24; from now on fits into int32 unchanged */
+	/* |h0| <= 1.8*2^37 */
+
+	carry[0] = (h0 + (1 << 25)) >> 26
+	h1 += carry[0]
+	h0 -= carry[0] << 26
+	/* |h0| <= 2^25; from now on fits into int32 unchanged */
+	/* |h1| <= 1.01*2^24 */
+
+	h[0] = int32(h0)
+	h[1] = int32(h1)
+	h[2] = int32(h2)
+	h[3] = int32(h3)
+	h[4] = int32(h4)
+	h[5] = int32(h5)
+	h[6] = int32(h6)
+	h[7] = int32(h7)
+	h[8] = int32(h8)
+	h[9] = int32(h9)
+}
+
+// feSquare calculates h = f*f. Can overlap h with f.
+//
+// Preconditions:
+//    |f| bounded by 1.1*2^26,1.1*2^25,1.1*2^26,1.1*2^25,etc.
+//
+// Postconditions:
+//    |h| bounded by 1.1*2^25,1.1*2^24,1.1*2^25,1.1*2^24,etc.
+func feSquare(h, f *fieldElement) {
+	f0 := f[0]
+	f1 := f[1]
+	f2 := f[2]
+	f3 := f[3]
+	f4 := f[4]
+	f5 := f[5]
+	f6 := f[6]
+	f7 := f[7]
+	f8 := f[8]
+	f9 := f[9]
+	f0_2 := 2 * f0
+	f1_2 := 2 * f1
+	f2_2 := 2 * f2
+	f3_2 := 2 * f3
+	f4_2 := 2 * f4
+	f5_2 := 2 * f5
+	f6_2 := 2 * f6
+	f7_2 := 2 * f7
+	f5_38 := 38 * f5 // 1.31*2^30
+	f6_19 := 19 * f6 // 1.31*2^30
+	f7_38 := 38 * f7 // 1.31*2^30
+	f8_19 := 19 * f8 // 1.31*2^30
+	f9_38 := 38 * f9 // 1.31*2^30
+	f0f0 := int64(f0) * int64(f0)
+	f0f1_2 := int64(f0_2) * int64(f1)
+	f0f2_2 := int64(f0_2) * int64(f2)
+	f0f3_2 := int64(f0_2) * int64(f3)
+	f0f4_2 := int64(f0_2) * int64(f4)
+	f0f5_2 := int64(f0_2) * int64(f5)
+	f0f6_2 := int64(f0_2) * int64(f6)
+	f0f7_2 := int64(f0_2) * int64(f7)
+	f0f8_2 := int64(f0_2) * int64(f8)
+	f0f9_2 := int64(f0_2) * int64(f9)
+	f1f1_2 := int64(f1_2) * int64(f1)
+	f1f2_2 := int64(f1_2) * int64(f2)
+	f1f3_4 := int64(f1_2) * int64(f3_2)
+	f1f4_2 := int64(f1_2) * int64(f4)
+	f1f5_4 := int64(f1_2) * int64(f5_2)
+	f1f6_2 := int64(f1_2) * int64(f6)
+	f1f7_4 := int64(f1_2) * int64(f7_2)
+	f1f8_2 := int64(f1_2) * int64(f8)
+	f1f9_76 := int64(f1_2) * int64(f9_38)
+	f2f2 := int64(f2) * int64(f2)
+	f2f3_2 := int64(f2_2) * int64(f3)
+	f2f4_2 := int64(f2_2) * int64(f4)
+	f2f5_2 := int64(f2_2) * int64(f5)
+	f2f6_2 := int64(f2_2) * int64(f6)
+	f2f7_2 := int64(f2_2) * int64(f7)
+	f2f8_38 := int64(f2_2) * int64(f8_19)
+	f2f9_38 := int64(f2) * int64(f9_38)
+	f3f3_2 := int64(f3_2) * int64(f3)
+	f3f4_2 := int64(f3_2) * int64(f4)
+	f3f5_4 := int64(f3_2) * int64(f5_2)
+	f3f6_2 := int64(f3_2) * int64(f6)
+	f3f7_76 := int64(f3_2) * int64(f7_38)
+	f3f8_38 := int64(f3_2) * int64(f8_19)
+	f3f9_76 := int64(f3_2) * int64(f9_38)
+	f4f4 := int64(f4) * int64(f4)
+	f4f5_2 := int64(f4_2) * int64(f5)
+	f4f6_38 := int64(f4_2) * int64(f6_19)
+	f4f7_38 := int64(f4) * int64(f7_38)
+	f4f8_38 := int64(f4_2) * int64(f8_19)
+	f4f9_38 := int64(f4) * int64(f9_38)
+	f5f5_38 := int64(f5) * int64(f5_38)
+	f5f6_38 := int64(f5_2) * int64(f6_19)
+	f5f7_76 := int64(f5_2) * int64(f7_38)
+	f5f8_38 := int64(f5_2) * int64(f8_19)
+	f5f9_76 := int64(f5_2) * int64(f9_38)
+	f6f6_19 := int64(f6) * int64(f6_19)
+	f6f7_38 := int64(f6) * int64(f7_38)
+	f6f8_38 := int64(f6_2) * int64(f8_19)
+	f6f9_38 := int64(f6) * int64(f9_38)
+	f7f7_38 := int64(f7) * int64(f7_38)
+	f7f8_38 := int64(f7_2) * int64(f8_19)
+	f7f9_76 := int64(f7_2) * int64(f9_38)
+	f8f8_19 := int64(f8) * int64(f8_19)
+	f8f9_38 := int64(f8) * int64(f9_38)
+	f9f9_38 := int64(f9) * int64(f9_38)
+	h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38
+	h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38
+	h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19
+	h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38
+	h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38
+	h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38
+	h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19
+	h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38
+	h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38
+	h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2
+	var carry [10]int64
+
+	carry[0] = (h0 + (1 << 25)) >> 26
+	h1 += carry[0]
+	h0 -= carry[0] << 26
+	carry[4] = (h4 + (1 << 25)) >> 26
+	h5 += carry[4]
+	h4 -= carry[4] << 26
+
+	carry[1] = (h1 + (1 << 24)) >> 25
+	h2 += carry[1]
+	h1 -= carry[1] << 25
+	carry[5] = (h5 + (1 << 24)) >> 25
+	h6 += carry[5]
+	h5 -= carry[5] << 25
+
+	carry[2] = (h2 + (1 << 25)) >> 26
+	h3 += carry[2]
+	h2 -= carry[2] << 26
+	carry[6] = (h6 + (1 << 25)) >> 26
+	h7 += carry[6]
+	h6 -= carry[6] << 26
+
+	carry[3] = (h3 + (1 << 24)) >> 25
+	h4 += carry[3]
+	h3 -= carry[3] << 25
+	carry[7] = (h7 + (1 << 24)) >> 25
+	h8 += carry[7]
+	h7 -= carry[7] << 25
+
+	carry[4] = (h4 + (1 << 25)) >> 26
+	h5 += carry[4]
+	h4 -= carry[4] << 26
+	carry[8] = (h8 + (1 << 25)) >> 26
+	h9 += carry[8]
+	h8 -= carry[8] << 26
+
+	carry[9] = (h9 + (1 << 24)) >> 25
+	h0 += carry[9] * 19
+	h9 -= carry[9] << 25
+
+	carry[0] = (h0 + (1 << 25)) >> 26
+	h1 += carry[0]
+	h0 -= carry[0] << 26
+
+	h[0] = int32(h0)
+	h[1] = int32(h1)
+	h[2] = int32(h2)
+	h[3] = int32(h3)
+	h[4] = int32(h4)
+	h[5] = int32(h5)
+	h[6] = int32(h6)
+	h[7] = int32(h7)
+	h[8] = int32(h8)
+	h[9] = int32(h9)
+}
+
+// feSquare2 sets h = 2 * f * f
+//
+// Can overlap h with f.
+//
+// Preconditions:
+//    |f| bounded by 1.65*2^26,1.65*2^25,1.65*2^26,1.65*2^25,etc.
+//
+// Postconditions:
+//    |h| bounded by 1.01*2^25,1.01*2^24,1.01*2^25,1.01*2^24,etc.
+// See fe_mul.c for discussion of implementation strategy.
+func feSquare2(h, f *fieldElement) {
+	f0 := f[0]
+	f1 := f[1]
+	f2 := f[2]
+	f3 := f[3]
+	f4 := f[4]
+	f5 := f[5]
+	f6 := f[6]
+	f7 := f[7]
+	f8 := f[8]
+	f9 := f[9]
+	f0_2 := 2 * f0
+	f1_2 := 2 * f1
+	f2_2 := 2 * f2
+	f3_2 := 2 * f3
+	f4_2 := 2 * f4
+	f5_2 := 2 * f5
+	f6_2 := 2 * f6
+	f7_2 := 2 * f7
+	f5_38 := 38 * f5 // 1.959375*2^30
+	f6_19 := 19 * f6 // 1.959375*2^30
+	f7_38 := 38 * f7 // 1.959375*2^30
+	f8_19 := 19 * f8 // 1.959375*2^30
+	f9_38 := 38 * f9 // 1.959375*2^30
+	f0f0 := int64(f0) * int64(f0)
+	f0f1_2 := int64(f0_2) * int64(f1)
+	f0f2_2 := int64(f0_2) * int64(f2)
+	f0f3_2 := int64(f0_2) * int64(f3)
+	f0f4_2 := int64(f0_2) * int64(f4)
+	f0f5_2 := int64(f0_2) * int64(f5)
+	f0f6_2 := int64(f0_2) * int64(f6)
+	f0f7_2 := int64(f0_2) * int64(f7)
+	f0f8_2 := int64(f0_2) * int64(f8)
+	f0f9_2 := int64(f0_2) * int64(f9)
+	f1f1_2 := int64(f1_2) * int64(f1)
+	f1f2_2 := int64(f1_2) * int64(f2)
+	f1f3_4 := int64(f1_2) * int64(f3_2)
+	f1f4_2 := int64(f1_2) * int64(f4)
+	f1f5_4 := int64(f1_2) * int64(f5_2)
+	f1f6_2 := int64(f1_2) * int64(f6)
+	f1f7_4 := int64(f1_2) * int64(f7_2)
+	f1f8_2 := int64(f1_2) * int64(f8)
+	f1f9_76 := int64(f1_2) * int64(f9_38)
+	f2f2 := int64(f2) * int64(f2)
+	f2f3_2 := int64(f2_2) * int64(f3)
+	f2f4_2 := int64(f2_2) * int64(f4)
+	f2f5_2 := int64(f2_2) * int64(f5)
+	f2f6_2 := int64(f2_2) * int64(f6)
+	f2f7_2 := int64(f2_2) * int64(f7)
+	f2f8_38 := int64(f2_2) * int64(f8_19)
+	f2f9_38 := int64(f2) * int64(f9_38)
+	f3f3_2 := int64(f3_2) * int64(f3)
+	f3f4_2 := int64(f3_2) * int64(f4)
+	f3f5_4 := int64(f3_2) * int64(f5_2)
+	f3f6_2 := int64(f3_2) * int64(f6)
+	f3f7_76 := int64(f3_2) * int64(f7_38)
+	f3f8_38 := int64(f3_2) * int64(f8_19)
+	f3f9_76 := int64(f3_2) * int64(f9_38)
+	f4f4 := int64(f4) * int64(f4)
+	f4f5_2 := int64(f4_2) * int64(f5)
+	f4f6_38 := int64(f4_2) * int64(f6_19)
+	f4f7_38 := int64(f4) * int64(f7_38)
+	f4f8_38 := int64(f4_2) * int64(f8_19)
+	f4f9_38 := int64(f4) * int64(f9_38)
+	f5f5_38 := int64(f5) * int64(f5_38)
+	f5f6_38 := int64(f5_2) * int64(f6_19)
+	f5f7_76 := int64(f5_2) * int64(f7_38)
+	f5f8_38 := int64(f5_2) * int64(f8_19)
+	f5f9_76 := int64(f5_2) * int64(f9_38)
+	f6f6_19 := int64(f6) * int64(f6_19)
+	f6f7_38 := int64(f6) * int64(f7_38)
+	f6f8_38 := int64(f6_2) * int64(f8_19)
+	f6f9_38 := int64(f6) * int64(f9_38)
+	f7f7_38 := int64(f7) * int64(f7_38)
+	f7f8_38 := int64(f7_2) * int64(f8_19)
+	f7f9_76 := int64(f7_2) * int64(f9_38)
+	f8f8_19 := int64(f8) * int64(f8_19)
+	f8f9_38 := int64(f8) * int64(f9_38)
+	f9f9_38 := int64(f9) * int64(f9_38)
+	h0 := f0f0 + f1f9_76 + f2f8_38 + f3f7_76 + f4f6_38 + f5f5_38
+	h1 := f0f1_2 + f2f9_38 + f3f8_38 + f4f7_38 + f5f6_38
+	h2 := f0f2_2 + f1f1_2 + f3f9_76 + f4f8_38 + f5f7_76 + f6f6_19
+	h3 := f0f3_2 + f1f2_2 + f4f9_38 + f5f8_38 + f6f7_38
+	h4 := f0f4_2 + f1f3_4 + f2f2 + f5f9_76 + f6f8_38 + f7f7_38
+	h5 := f0f5_2 + f1f4_2 + f2f3_2 + f6f9_38 + f7f8_38
+	h6 := f0f6_2 + f1f5_4 + f2f4_2 + f3f3_2 + f7f9_76 + f8f8_19
+	h7 := f0f7_2 + f1f6_2 + f2f5_2 + f3f4_2 + f8f9_38
+	h8 := f0f8_2 + f1f7_4 + f2f6_2 + f3f5_4 + f4f4 + f9f9_38
+	h9 := f0f9_2 + f1f8_2 + f2f7_2 + f3f6_2 + f4f5_2
+	var carry [10]int64
+
+	h0 += h0
+	h1 += h1
+	h2 += h2
+	h3 += h3
+	h4 += h4
+	h5 += h5
+	h6 += h6
+	h7 += h7
+	h8 += h8
+	h9 += h9
+
+	carry[0] = (h0 + (1 << 25)) >> 26
+	h1 += carry[0]
+	h0 -= carry[0] << 26
+	carry[4] = (h4 + (1 << 25)) >> 26
+	h5 += carry[4]
+	h4 -= carry[4] << 26
+
+	carry[1] = (h1 + (1 << 24)) >> 25
+	h2 += carry[1]
+	h1 -= carry[1] << 25
+	carry[5] = (h5 + (1 << 24)) >> 25
+	h6 += carry[5]
+	h5 -= carry[5] << 25
+
+	carry[2] = (h2 + (1 << 25)) >> 26
+	h3 += carry[2]
+	h2 -= carry[2] << 26
+	carry[6] = (h6 + (1 << 25)) >> 26
+	h7 += carry[6]
+	h6 -= carry[6] << 26
+
+	carry[3] = (h3 + (1 << 24)) >> 25
+	h4 += carry[3]
+	h3 -= carry[3] << 25
+	carry[7] = (h7 + (1 << 24)) >> 25
+	h8 += carry[7]
+	h7 -= carry[7] << 25
+
+	carry[4] = (h4 + (1 << 25)) >> 26
+	h5 += carry[4]
+	h4 -= carry[4] << 26
+	carry[8] = (h8 + (1 << 25)) >> 26
+	h9 += carry[8]
+	h8 -= carry[8] << 26
+
+	carry[9] = (h9 + (1 << 24)) >> 25
+	h0 += carry[9] * 19
+	h9 -= carry[9] << 25
+
+	carry[0] = (h0 + (1 << 25)) >> 26
+	h1 += carry[0]
+	h0 -= carry[0] << 26
+
+	h[0] = int32(h0)
+	h[1] = int32(h1)
+	h[2] = int32(h2)
+	h[3] = int32(h3)
+	h[4] = int32(h4)
+	h[5] = int32(h5)
+	h[6] = int32(h6)
+	h[7] = int32(h7)
+	h[8] = int32(h8)
+	h[9] = int32(h9)
+}
+
+func feInvert(out, z *fieldElement) {
+	var t0, t1, t2, t3 fieldElement
+	var i int
+
+	feSquare(&t0, z)
+	for i = 1; i < 1; i++ {
+		feSquare(&t0, &t0)
+	}
+	feSquare(&t1, &t0)
+	for i = 1; i < 2; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(&t1, z, &t1)
+	feMul(&t0, &t0, &t1)
+	feSquare(&t2, &t0)
+	for i = 1; i < 1; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t1, &t1, &t2)
+	feSquare(&t2, &t1)
+	for i = 1; i < 5; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t1, &t2, &t1)
+	feSquare(&t2, &t1)
+	for i = 1; i < 10; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t2, &t2, &t1)
+	feSquare(&t3, &t2)
+	for i = 1; i < 20; i++ {
+		feSquare(&t3, &t3)
+	}
+	feMul(&t2, &t3, &t2)
+	feSquare(&t2, &t2)
+	for i = 1; i < 10; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t1, &t2, &t1)
+	feSquare(&t2, &t1)
+	for i = 1; i < 50; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t2, &t2, &t1)
+	feSquare(&t3, &t2)
+	for i = 1; i < 100; i++ {
+		feSquare(&t3, &t3)
+	}
+	feMul(&t2, &t3, &t2)
+	feSquare(&t2, &t2)
+	for i = 1; i < 50; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t1, &t2, &t1)
+	feSquare(&t1, &t1)
+	for i = 1; i < 5; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(out, &t1, &t0)
+}
+
+func fePow22523(out, z *fieldElement) {
+	var t0, t1, t2 fieldElement
+	var i int
+
+	feSquare(&t0, z)
+	for i = 1; i < 1; i++ {
+		feSquare(&t0, &t0)
+	}
+	feSquare(&t1, &t0)
+	for i = 1; i < 2; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(&t1, z, &t1)
+	feMul(&t0, &t0, &t1)
+	feSquare(&t0, &t0)
+	for i = 1; i < 1; i++ {
+		feSquare(&t0, &t0)
+	}
+	feMul(&t0, &t1, &t0)
+	feSquare(&t1, &t0)
+	for i = 1; i < 5; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(&t0, &t1, &t0)
+	feSquare(&t1, &t0)
+	for i = 1; i < 10; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(&t1, &t1, &t0)
+	feSquare(&t2, &t1)
+	for i = 1; i < 20; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t1, &t2, &t1)
+	feSquare(&t1, &t1)
+	for i = 1; i < 10; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(&t0, &t1, &t0)
+	feSquare(&t1, &t0)
+	for i = 1; i < 50; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(&t1, &t1, &t0)
+	feSquare(&t2, &t1)
+	for i = 1; i < 100; i++ {
+		feSquare(&t2, &t2)
+	}
+	feMul(&t1, &t2, &t1)
+	feSquare(&t1, &t1)
+	for i = 1; i < 50; i++ {
+		feSquare(&t1, &t1)
+	}
+	feMul(&t0, &t1, &t0)
+	feSquare(&t0, &t0)
+	for i = 1; i < 2; i++ {
+		feSquare(&t0, &t0)
+	}
+	feMul(out, &t0, z)
+}
+
+// Group elements are members of the elliptic curve -x^2 + y^2 = 1 + d * x^2 *
+// y^2 where d = -121665/121666.
+//
+// Several representations are used:
+//   projectiveGroupElement: (X:Y:Z) satisfying x=X/Z, y=Y/Z
+//   extendedGroupElement: (X:Y:Z:T) satisfying x=X/Z, y=Y/Z, XY=ZT
+//   completedGroupElement: ((X:Z),(Y:T)) satisfying x=X/Z, y=Y/T
+//   preComputedGroupElement: (y+x,y-x,2dxy)
+
+type projectiveGroupElement struct {
+	X, Y, Z fieldElement
+}
+
+type extendedGroupElement struct {
+	X, Y, Z, T fieldElement
+}
+
+type completedGroupElement struct {
+	X, Y, Z, T fieldElement
+}
+
+type preComputedGroupElement struct {
+	yPlusX, yMinusX, xy2d fieldElement
+}
+
+type cachedGroupElement struct {
+	yPlusX, yMinusX, Z, T2d fieldElement
+}
+
+func (p *projectiveGroupElement) Zero() {
+	feZero(&p.X)
+	feOne(&p.Y)
+	feOne(&p.Z)
+}
+
+func (p *projectiveGroupElement) Double(r *completedGroupElement) {
+	var t0 fieldElement
+
+	feSquare(&r.X, &p.X)
+	feSquare(&r.Z, &p.Y)
+	feSquare2(&r.T, &p.Z)
+	feAdd(&r.Y, &p.X, &p.Y)
+	feSquare(&t0, &r.Y)
+	feAdd(&r.Y, &r.Z, &r.X)
+	feSub(&r.Z, &r.Z, &r.X)
+	feSub(&r.X, &t0, &r.Y)
+	feSub(&r.T, &r.T, &r.Z)
+}
+
+func (p *projectiveGroupElement) ToBytes(s *[32]byte) {
+	var recip, x, y fieldElement
+
+	feInvert(&recip, &p.Z)
+	feMul(&x, &p.X, &recip)
+	feMul(&y, &p.Y, &recip)
+	feToBytes(s, &y)
+	s[31] ^= feIsNegative(&x) << 7
+}
+
+func (p *extendedGroupElement) Zero() {
+	feZero(&p.X)
+	feOne(&p.Y)
+	feOne(&p.Z)
+	feZero(&p.T)
+}
+
+func (p *extendedGroupElement) Double(r *completedGroupElement) {
+	var q projectiveGroupElement
+	p.ToProjective(&q)
+	q.Double(r)
+}
+
+func (p *extendedGroupElement) ToCached(r *cachedGroupElement) {
+	feAdd(&r.yPlusX, &p.Y, &p.X)
+	feSub(&r.yMinusX, &p.Y, &p.X)
+	feCopy(&r.Z, &p.Z)
+	feMul(&r.T2d, &p.T, &d2)
+}
+
+func (p *extendedGroupElement) ToProjective(r *projectiveGroupElement) {
+	feCopy(&r.X, &p.X)
+	feCopy(&r.Y, &p.Y)
+	feCopy(&r.Z, &p.Z)
+}
+
+func (p *extendedGroupElement) ToBytes(s *[32]byte) {
+	var recip, x, y fieldElement
+
+	feInvert(&recip, &p.Z)
+	feMul(&x, &p.X, &recip)
+	feMul(&y, &p.Y, &recip)
+	feToBytes(s, &y)
+	s[31] ^= feIsNegative(&x) << 7
+}
+
+func (p *extendedGroupElement) FromBytes(s *[32]byte) bool {
+	var u, v, v3, vxx, check fieldElement
+
+	feFromBytes(&p.Y, s)
+	feOne(&p.Z)
+	feSquare(&u, &p.Y)
+	feMul(&v, &u, &d)
+	feSub(&u, &u, &p.Z) // y = y^2-1
+	feAdd(&v, &v, &p.Z) // v = dy^2+1
+
+	feSquare(&v3, &v)
+	feMul(&v3, &v3, &v) // v3 = v^3
+	feSquare(&p.X, &v3)
+	feMul(&p.X, &p.X, &v)
+	feMul(&p.X, &p.X, &u) // x = uv^7
+
+	fePow22523(&p.X, &p.X) // x = (uv^7)^((q-5)/8)
+	feMul(&p.X, &p.X, &v3)
+	feMul(&p.X, &p.X, &u) // x = uv^3(uv^7)^((q-5)/8)
+
+	var tmpX, tmp2 [32]byte
+
+	feSquare(&vxx, &p.X)
+	feMul(&vxx, &vxx, &v)
+	feSub(&check, &vxx, &u) // vx^2-u
+	if feIsNonZero(&check) {
+		feAdd(&check, &vxx, &u) // vx^2+u
+		if feIsNonZero(&check) {
+			return false
+		}
+		feMul(&p.X, &p.X, &sqrtm1)
+
+		feToBytes(&tmpX, &p.X)
+		for i, v := range tmpX {
+			tmp2[31-i] = v
+		}
+	}
+
+	if feIsNegative(&p.X) == (s[31] >> 7) {
+		feNeg(&p.X, &p.X)
+	}
+
+	feMul(&p.T, &p.X, &p.Y)
+	return true
+}
+
+func (p *completedGroupElement) ToProjective(r *projectiveGroupElement) {
+	feMul(&r.X, &p.X, &p.T)
+	feMul(&r.Y, &p.Y, &p.Z)
+	feMul(&r.Z, &p.Z, &p.T)
+}
+
+func (p *completedGroupElement) ToExtended(r *extendedGroupElement) {
+	feMul(&r.X, &p.X, &p.T)
+	feMul(&r.Y, &p.Y, &p.Z)
+	feMul(&r.Z, &p.Z, &p.T)
+	feMul(&r.T, &p.X, &p.Y)
+}
+
+func (p *preComputedGroupElement) Zero() {
+	feOne(&p.yPlusX)
+	feOne(&p.yMinusX)
+	feZero(&p.xy2d)
+}
+
+func geAdd(r *completedGroupElement, p *extendedGroupElement, q *cachedGroupElement) {
+	var t0 fieldElement
+
+	feAdd(&r.X, &p.Y, &p.X)
+	feSub(&r.Y, &p.Y, &p.X)
+	feMul(&r.Z, &r.X, &q.yPlusX)
+	feMul(&r.Y, &r.Y, &q.yMinusX)
+	feMul(&r.T, &q.T2d, &p.T)
+	feMul(&r.X, &p.Z, &q.Z)
+	feAdd(&t0, &r.X, &r.X)
+	feSub(&r.X, &r.Z, &r.Y)
+	feAdd(&r.Y, &r.Z, &r.Y)
+	feAdd(&r.Z, &t0, &r.T)
+	feSub(&r.T, &t0, &r.T)
+}
+
+func geSub(r *completedGroupElement, p *extendedGroupElement, q *cachedGroupElement) {
+	var t0 fieldElement
+
+	feAdd(&r.X, &p.Y, &p.X)
+	feSub(&r.Y, &p.Y, &p.X)
+	feMul(&r.Z, &r.X, &q.yMinusX)
+	feMul(&r.Y, &r.Y, &q.yPlusX)
+	feMul(&r.T, &q.T2d, &p.T)
+	feMul(&r.X, &p.Z, &q.Z)
+	feAdd(&t0, &r.X, &r.X)
+	feSub(&r.X, &r.Z, &r.Y)
+	feAdd(&r.Y, &r.Z, &r.Y)
+	feSub(&r.Z, &t0, &r.T)
+	feAdd(&r.T, &t0, &r.T)
+}
+
+func geMixedAdd(r *completedGroupElement, p *extendedGroupElement, q *preComputedGroupElement) {
+	var t0 fieldElement
+
+	feAdd(&r.X, &p.Y, &p.X)
+	feSub(&r.Y, &p.Y, &p.X)
+	feMul(&r.Z, &r.X, &q.yPlusX)
+	feMul(&r.Y, &r.Y, &q.yMinusX)
+	feMul(&r.T, &q.xy2d, &p.T)
+	feAdd(&t0, &p.Z, &p.Z)
+	feSub(&r.X, &r.Z, &r.Y)
+	feAdd(&r.Y, &r.Z, &r.Y)
+	feAdd(&r.Z, &t0, &r.T)
+	feSub(&r.T, &t0, &r.T)
+}
+
+func geMixedSub(r *completedGroupElement, p *extendedGroupElement, q *preComputedGroupElement) {
+	var t0 fieldElement
+
+	feAdd(&r.X, &p.Y, &p.X)
+	feSub(&r.Y, &p.Y, &p.X)
+	feMul(&r.Z, &r.X, &q.yMinusX)
+	feMul(&r.Y, &r.Y, &q.yPlusX)
+	feMul(&r.T, &q.xy2d, &p.T)
+	feAdd(&t0, &p.Z, &p.Z)
+	feSub(&r.X, &r.Z, &r.Y)
+	feAdd(&r.Y, &r.Z, &r.Y)
+	feSub(&r.Z, &t0, &r.T)
+	feAdd(&r.T, &t0, &r.T)
+}
+
+func slide(r *[256]int8, a *[32]byte) {
+	for i := range r {
+		r[i] = int8(1 & (a[i>>3] >> uint(i&7)))
+	}
+
+	for i := range r {
+		if r[i] != 0 {
+			for b := 1; b <= 6 && i+b < 256; b++ {
+				if r[i+b] != 0 {
+					if r[i]+(r[i+b]<<uint(b)) <= 15 {
+						r[i] += r[i+b] << uint(b)
+						r[i+b] = 0
+					} else if r[i]-(r[i+b]<<uint(b)) >= -15 {
+						r[i] -= r[i+b] << uint(b)
+						for k := i + b; k < 256; k++ {
+							if r[k] == 0 {
+								r[k] = 1
+								break
+							}
+							r[k] = 0
+						}
+					} else {
+						break
+					}
+				}
+			}
+		}
+	}
+}
+
+// geDoubleScalarMultVartime sets r = a*A + b*B
+// where a = a[0]+256*a[1]+...+256^31 a[31].
+// and b = b[0]+256*b[1]+...+256^31 b[31].
+// B is the Ed25519 base point (x,4/5) with x positive.
+func geDoubleScalarMultVartime(r *projectiveGroupElement, a *[32]byte, A *extendedGroupElement, b *[32]byte) {
+	var aSlide, bSlide [256]int8
+	var Ai [8]cachedGroupElement // A,3A,5A,7A,9A,11A,13A,15A
+	var t completedGroupElement
+	var u, A2 extendedGroupElement
+	var i int
+
+	slide(&aSlide, a)
+	slide(&bSlide, b)
+
+	A.ToCached(&Ai[0])
+	A.Double(&t)
+	t.ToExtended(&A2)
+
+	for i := 0; i < 7; i++ {
+		geAdd(&t, &A2, &Ai[i])
+		t.ToExtended(&u)
+		u.ToCached(&Ai[i+1])
+	}
+
+	r.Zero()
+
+	for i = 255; i >= 0; i-- {
+		if aSlide[i] != 0 || bSlide[i] != 0 {
+			break
+		}
+	}
+
+	for ; i >= 0; i-- {
+		r.Double(&t)
+
+		if aSlide[i] > 0 {
+			t.ToExtended(&u)
+			geAdd(&t, &u, &Ai[aSlide[i]/2])
+		} else if aSlide[i] < 0 {
+			t.ToExtended(&u)
+			geSub(&t, &u, &Ai[(-aSlide[i])/2])
+		}
+
+		if bSlide[i] > 0 {
+			t.ToExtended(&u)
+			geMixedAdd(&t, &u, &bi[bSlide[i]/2])
+		} else if bSlide[i] < 0 {
+			t.ToExtended(&u)
+			geMixedSub(&t, &u, &bi[(-bSlide[i])/2])
+		}
+
+		t.ToProjective(r)
+	}
+}
+
+// equal returns 1 if b == c and 0 otherwise.
+func equal(b, c int32) int32 {
+	x := uint32(b ^ c)
+	x--
+	return int32(x >> 31)
+}
+
+// negative returns 1 if b < 0 and 0 otherwise.
+func negative(b int32) int32 {
+	return (b >> 31) & 1
+}
+
+func preComputedGroupElementCMove(t, u *preComputedGroupElement, b int32) {
+	feCMove(&t.yPlusX, &u.yPlusX, b)
+	feCMove(&t.yMinusX, &u.yMinusX, b)
+	feCMove(&t.xy2d, &u.xy2d, b)
+}
+
+func selectPoint(t *preComputedGroupElement, pos int32, b int32) {
+	var minusT preComputedGroupElement
+	bNegative := negative(b)
+	bAbs := b - (((-bNegative) & b) << 1)
+
+	t.Zero()
+	for i := int32(0); i < 8; i++ {
+		preComputedGroupElementCMove(t, &base[pos][i], equal(bAbs, i+1))
+	}
+	feCopy(&minusT.yPlusX, &t.yMinusX)
+	feCopy(&minusT.yMinusX, &t.yPlusX)
+	feNeg(&minusT.xy2d, &t.xy2d)
+	preComputedGroupElementCMove(t, &minusT, bNegative)
+}
+
+// geScalarMultBase computes h = a*B, where
+//   a = a[0]+256*a[1]+...+256^31 a[31]
+//   B is the Ed25519 base point (x,4/5) with x positive.
+//
+// Preconditions:
+//   a[31] <= 127
+func geScalarMultBase(h *extendedGroupElement, a *[32]byte) {
+	var e [64]int8
+
+	for i, v := range a {
+		e[2*i] = int8(v & 15)
+		e[2*i+1] = int8((v >> 4) & 15)
+	}
+
+	// each e[i] is between 0 and 15 and e[63] is between 0 and 7.
+
+	carry := int8(0)
+	for i := 0; i < 63; i++ {
+		e[i] += carry
+		carry = (e[i] + 8) >> 4
+		e[i] -= carry << 4
+	}
+	e[63] += carry
+	// each e[i] is between -8 and 8.
+
+	h.Zero()
+	var t preComputedGroupElement
+	var r completedGroupElement
+	for i := int32(1); i < 64; i += 2 {
+		selectPoint(&t, i/2, int32(e[i]))
+		geMixedAdd(&r, h, &t)
+		r.ToExtended(h)
+	}
+
+	var s projectiveGroupElement
+
+	h.Double(&r)
+	r.ToProjective(&s)
+	s.Double(&r)
+	r.ToProjective(&s)
+	s.Double(&r)
+	r.ToProjective(&s)
+	s.Double(&r)
+	r.ToExtended(h)
+
+	for i := int32(0); i < 64; i += 2 {
+		selectPoint(&t, i/2, int32(e[i]))
+		geMixedAdd(&r, h, &t)
+		r.ToExtended(h)
+	}
+}
+
+// The scalars are GF(2^252 + 27742317777372353535851937790883648493).
+
+// Input:
+//   a[0]+256*a[1]+...+256^31*a[31] = a
+//   b[0]+256*b[1]+...+256^31*b[31] = b
+//   c[0]+256*c[1]+...+256^31*c[31] = c
+//
+// Output:
+//   s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l
+//   where l = 2^252 + 27742317777372353535851937790883648493.
+func scMulAdd(s, a, b, c *[32]byte) {
+	a0 := 2097151 & load3(a[:])
+	a1 := 2097151 & (load4(a[2:]) >> 5)
+	a2 := 2097151 & (load3(a[5:]) >> 2)
+	a3 := 2097151 & (load4(a[7:]) >> 7)
+	a4 := 2097151 & (load4(a[10:]) >> 4)
+	a5 := 2097151 & (load3(a[13:]) >> 1)
+	a6 := 2097151 & (load4(a[15:]) >> 6)
+	a7 := 2097151 & (load3(a[18:]) >> 3)
+	a8 := 2097151 & load3(a[21:])
+	a9 := 2097151 & (load4(a[23:]) >> 5)
+	a10 := 2097151 & (load3(a[26:]) >> 2)
+	a11 := (load4(a[28:]) >> 7)
+	b0 := 2097151 & load3(b[:])
+	b1 := 2097151 & (load4(b[2:]) >> 5)
+	b2 := 2097151 & (load3(b[5:]) >> 2)
+	b3 := 2097151 & (load4(b[7:]) >> 7)
+	b4 := 2097151 & (load4(b[10:]) >> 4)
+	b5 := 2097151 & (load3(b[13:]) >> 1)
+	b6 := 2097151 & (load4(b[15:]) >> 6)
+	b7 := 2097151 & (load3(b[18:]) >> 3)
+	b8 := 2097151 & load3(b[21:])
+	b9 := 2097151 & (load4(b[23:]) >> 5)
+	b10 := 2097151 & (load3(b[26:]) >> 2)
+	b11 := (load4(b[28:]) >> 7)
+	c0 := 2097151 & load3(c[:])
+	c1 := 2097151 & (load4(c[2:]) >> 5)
+	c2 := 2097151 & (load3(c[5:]) >> 2)
+	c3 := 2097151 & (load4(c[7:]) >> 7)
+	c4 := 2097151 & (load4(c[10:]) >> 4)
+	c5 := 2097151 & (load3(c[13:]) >> 1)
+	c6 := 2097151 & (load4(c[15:]) >> 6)
+	c7 := 2097151 & (load3(c[18:]) >> 3)
+	c8 := 2097151 & load3(c[21:])
+	c9 := 2097151 & (load4(c[23:]) >> 5)
+	c10 := 2097151 & (load3(c[26:]) >> 2)
+	c11 := (load4(c[28:]) >> 7)
+	var carry [23]int64
+
+	s0 := c0 + a0*b0
+	s1 := c1 + a0*b1 + a1*b0
+	s2 := c2 + a0*b2 + a1*b1 + a2*b0
+	s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0
+	s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0
+	s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0
+	s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0
+	s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0
+	s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0
+	s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0
+	s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0
+	s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0
+	s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1
+	s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2
+	s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3
+	s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4
+	s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5
+	s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6
+	s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7
+	s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8
+	s20 := a9*b11 + a10*b10 + a11*b9
+	s21 := a10*b11 + a11*b10
+	s22 := a11 * b11
+	s23 := int64(0)
+
+	carry[0] = (s0 + (1 << 20)) >> 21
+	s1 += carry[0]
+	s0 -= carry[0] << 21
+	carry[2] = (s2 + (1 << 20)) >> 21
+	s3 += carry[2]
+	s2 -= carry[2] << 21
+	carry[4] = (s4 + (1 << 20)) >> 21
+	s5 += carry[4]
+	s4 -= carry[4] << 21
+	carry[6] = (s6 + (1 << 20)) >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[8] = (s8 + (1 << 20)) >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[10] = (s10 + (1 << 20)) >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+	carry[12] = (s12 + (1 << 20)) >> 21
+	s13 += carry[12]
+	s12 -= carry[12] << 21
+	carry[14] = (s14 + (1 << 20)) >> 21
+	s15 += carry[14]
+	s14 -= carry[14] << 21
+	carry[16] = (s16 + (1 << 20)) >> 21
+	s17 += carry[16]
+	s16 -= carry[16] << 21
+	carry[18] = (s18 + (1 << 20)) >> 21
+	s19 += carry[18]
+	s18 -= carry[18] << 21
+	carry[20] = (s20 + (1 << 20)) >> 21
+	s21 += carry[20]
+	s20 -= carry[20] << 21
+	carry[22] = (s22 + (1 << 20)) >> 21
+	s23 += carry[22]
+	s22 -= carry[22] << 21
+
+	carry[1] = (s1 + (1 << 20)) >> 21
+	s2 += carry[1]
+	s1 -= carry[1] << 21
+	carry[3] = (s3 + (1 << 20)) >> 21
+	s4 += carry[3]
+	s3 -= carry[3] << 21
+	carry[5] = (s5 + (1 << 20)) >> 21
+	s6 += carry[5]
+	s5 -= carry[5] << 21
+	carry[7] = (s7 + (1 << 20)) >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[9] = (s9 + (1 << 20)) >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[11] = (s11 + (1 << 20)) >> 21
+	s12 += carry[11]
+	s11 -= carry[11] << 21
+	carry[13] = (s13 + (1 << 20)) >> 21
+	s14 += carry[13]
+	s13 -= carry[13] << 21
+	carry[15] = (s15 + (1 << 20)) >> 21
+	s16 += carry[15]
+	s15 -= carry[15] << 21
+	carry[17] = (s17 + (1 << 20)) >> 21
+	s18 += carry[17]
+	s17 -= carry[17] << 21
+	carry[19] = (s19 + (1 << 20)) >> 21
+	s20 += carry[19]
+	s19 -= carry[19] << 21
+	carry[21] = (s21 + (1 << 20)) >> 21
+	s22 += carry[21]
+	s21 -= carry[21] << 21
+
+	s11 += s23 * 666643
+	s12 += s23 * 470296
+	s13 += s23 * 654183
+	s14 -= s23 * 997805
+	s15 += s23 * 136657
+	s16 -= s23 * 683901
+	s23 = 0
+
+	s10 += s22 * 666643
+	s11 += s22 * 470296
+	s12 += s22 * 654183
+	s13 -= s22 * 997805
+	s14 += s22 * 136657
+	s15 -= s22 * 683901
+	s22 = 0
+
+	s9 += s21 * 666643
+	s10 += s21 * 470296
+	s11 += s21 * 654183
+	s12 -= s21 * 997805
+	s13 += s21 * 136657
+	s14 -= s21 * 683901
+	s21 = 0
+
+	s8 += s20 * 666643
+	s9 += s20 * 470296
+	s10 += s20 * 654183
+	s11 -= s20 * 997805
+	s12 += s20 * 136657
+	s13 -= s20 * 683901
+	s20 = 0
+
+	s7 += s19 * 666643
+	s8 += s19 * 470296
+	s9 += s19 * 654183
+	s10 -= s19 * 997805
+	s11 += s19 * 136657
+	s12 -= s19 * 683901
+	s19 = 0
+
+	s6 += s18 * 666643
+	s7 += s18 * 470296
+	s8 += s18 * 654183
+	s9 -= s18 * 997805
+	s10 += s18 * 136657
+	s11 -= s18 * 683901
+	s18 = 0
+
+	carry[6] = (s6 + (1 << 20)) >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[8] = (s8 + (1 << 20)) >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[10] = (s10 + (1 << 20)) >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+	carry[12] = (s12 + (1 << 20)) >> 21
+	s13 += carry[12]
+	s12 -= carry[12] << 21
+	carry[14] = (s14 + (1 << 20)) >> 21
+	s15 += carry[14]
+	s14 -= carry[14] << 21
+	carry[16] = (s16 + (1 << 20)) >> 21
+	s17 += carry[16]
+	s16 -= carry[16] << 21
+
+	carry[7] = (s7 + (1 << 20)) >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[9] = (s9 + (1 << 20)) >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[11] = (s11 + (1 << 20)) >> 21
+	s12 += carry[11]
+	s11 -= carry[11] << 21
+	carry[13] = (s13 + (1 << 20)) >> 21
+	s14 += carry[13]
+	s13 -= carry[13] << 21
+	carry[15] = (s15 + (1 << 20)) >> 21
+	s16 += carry[15]
+	s15 -= carry[15] << 21
+
+	s5 += s17 * 666643
+	s6 += s17 * 470296
+	s7 += s17 * 654183
+	s8 -= s17 * 997805
+	s9 += s17 * 136657
+	s10 -= s17 * 683901
+	s17 = 0
+
+	s4 += s16 * 666643
+	s5 += s16 * 470296
+	s6 += s16 * 654183
+	s7 -= s16 * 997805
+	s8 += s16 * 136657
+	s9 -= s16 * 683901
+	s16 = 0
+
+	s3 += s15 * 666643
+	s4 += s15 * 470296
+	s5 += s15 * 654183
+	s6 -= s15 * 997805
+	s7 += s15 * 136657
+	s8 -= s15 * 683901
+	s15 = 0
+
+	s2 += s14 * 666643
+	s3 += s14 * 470296
+	s4 += s14 * 654183
+	s5 -= s14 * 997805
+	s6 += s14 * 136657
+	s7 -= s14 * 683901
+	s14 = 0
+
+	s1 += s13 * 666643
+	s2 += s13 * 470296
+	s3 += s13 * 654183
+	s4 -= s13 * 997805
+	s5 += s13 * 136657
+	s6 -= s13 * 683901
+	s13 = 0
+
+	s0 += s12 * 666643
+	s1 += s12 * 470296
+	s2 += s12 * 654183
+	s3 -= s12 * 997805
+	s4 += s12 * 136657
+	s5 -= s12 * 683901
+	s12 = 0
+
+	carry[0] = (s0 + (1 << 20)) >> 21
+	s1 += carry[0]
+	s0 -= carry[0] << 21
+	carry[2] = (s2 + (1 << 20)) >> 21
+	s3 += carry[2]
+	s2 -= carry[2] << 21
+	carry[4] = (s4 + (1 << 20)) >> 21
+	s5 += carry[4]
+	s4 -= carry[4] << 21
+	carry[6] = (s6 + (1 << 20)) >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[8] = (s8 + (1 << 20)) >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[10] = (s10 + (1 << 20)) >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+
+	carry[1] = (s1 + (1 << 20)) >> 21
+	s2 += carry[1]
+	s1 -= carry[1] << 21
+	carry[3] = (s3 + (1 << 20)) >> 21
+	s4 += carry[3]
+	s3 -= carry[3] << 21
+	carry[5] = (s5 + (1 << 20)) >> 21
+	s6 += carry[5]
+	s5 -= carry[5] << 21
+	carry[7] = (s7 + (1 << 20)) >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[9] = (s9 + (1 << 20)) >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[11] = (s11 + (1 << 20)) >> 21
+	s12 += carry[11]
+	s11 -= carry[11] << 21
+
+	s0 += s12 * 666643
+	s1 += s12 * 470296
+	s2 += s12 * 654183
+	s3 -= s12 * 997805
+	s4 += s12 * 136657
+	s5 -= s12 * 683901
+	s12 = 0
+
+	carry[0] = s0 >> 21
+	s1 += carry[0]
+	s0 -= carry[0] << 21
+	carry[1] = s1 >> 21
+	s2 += carry[1]
+	s1 -= carry[1] << 21
+	carry[2] = s2 >> 21
+	s3 += carry[2]
+	s2 -= carry[2] << 21
+	carry[3] = s3 >> 21
+	s4 += carry[3]
+	s3 -= carry[3] << 21
+	carry[4] = s4 >> 21
+	s5 += carry[4]
+	s4 -= carry[4] << 21
+	carry[5] = s5 >> 21
+	s6 += carry[5]
+	s5 -= carry[5] << 21
+	carry[6] = s6 >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[7] = s7 >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[8] = s8 >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[9] = s9 >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[10] = s10 >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+	carry[11] = s11 >> 21
+	s12 += carry[11]
+	s11 -= carry[11] << 21
+
+	s0 += s12 * 666643
+	s1 += s12 * 470296
+	s2 += s12 * 654183
+	s3 -= s12 * 997805
+	s4 += s12 * 136657
+	s5 -= s12 * 683901
+	s12 = 0
+
+	carry[0] = s0 >> 21
+	s1 += carry[0]
+	s0 -= carry[0] << 21
+	carry[1] = s1 >> 21
+	s2 += carry[1]
+	s1 -= carry[1] << 21
+	carry[2] = s2 >> 21
+	s3 += carry[2]
+	s2 -= carry[2] << 21
+	carry[3] = s3 >> 21
+	s4 += carry[3]
+	s3 -= carry[3] << 21
+	carry[4] = s4 >> 21
+	s5 += carry[4]
+	s4 -= carry[4] << 21
+	carry[5] = s5 >> 21
+	s6 += carry[5]
+	s5 -= carry[5] << 21
+	carry[6] = s6 >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[7] = s7 >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[8] = s8 >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[9] = s9 >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[10] = s10 >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+
+	s[0] = byte(s0 >> 0)
+	s[1] = byte(s0 >> 8)
+	s[2] = byte((s0 >> 16) | (s1 << 5))
+	s[3] = byte(s1 >> 3)
+	s[4] = byte(s1 >> 11)
+	s[5] = byte((s1 >> 19) | (s2 << 2))
+	s[6] = byte(s2 >> 6)
+	s[7] = byte((s2 >> 14) | (s3 << 7))
+	s[8] = byte(s3 >> 1)
+	s[9] = byte(s3 >> 9)
+	s[10] = byte((s3 >> 17) | (s4 << 4))
+	s[11] = byte(s4 >> 4)
+	s[12] = byte(s4 >> 12)
+	s[13] = byte((s4 >> 20) | (s5 << 1))
+	s[14] = byte(s5 >> 7)
+	s[15] = byte((s5 >> 15) | (s6 << 6))
+	s[16] = byte(s6 >> 2)
+	s[17] = byte(s6 >> 10)
+	s[18] = byte((s6 >> 18) | (s7 << 3))
+	s[19] = byte(s7 >> 5)
+	s[20] = byte(s7 >> 13)
+	s[21] = byte(s8 >> 0)
+	s[22] = byte(s8 >> 8)
+	s[23] = byte((s8 >> 16) | (s9 << 5))
+	s[24] = byte(s9 >> 3)
+	s[25] = byte(s9 >> 11)
+	s[26] = byte((s9 >> 19) | (s10 << 2))
+	s[27] = byte(s10 >> 6)
+	s[28] = byte((s10 >> 14) | (s11 << 7))
+	s[29] = byte(s11 >> 1)
+	s[30] = byte(s11 >> 9)
+	s[31] = byte(s11 >> 17)
+}
+
+// Input:
+//   s[0]+256*s[1]+...+256^63*s[63] = s
+//
+// Output:
+//   s[0]+256*s[1]+...+256^31*s[31] = s mod l
+//   where l = 2^252 + 27742317777372353535851937790883648493.
+func scReduce(out *[32]byte, s *[64]byte) {
+	s0 := 2097151 & load3(s[:])
+	s1 := 2097151 & (load4(s[2:]) >> 5)
+	s2 := 2097151 & (load3(s[5:]) >> 2)
+	s3 := 2097151 & (load4(s[7:]) >> 7)
+	s4 := 2097151 & (load4(s[10:]) >> 4)
+	s5 := 2097151 & (load3(s[13:]) >> 1)
+	s6 := 2097151 & (load4(s[15:]) >> 6)
+	s7 := 2097151 & (load3(s[18:]) >> 3)
+	s8 := 2097151 & load3(s[21:])
+	s9 := 2097151 & (load4(s[23:]) >> 5)
+	s10 := 2097151 & (load3(s[26:]) >> 2)
+	s11 := 2097151 & (load4(s[28:]) >> 7)
+	s12 := 2097151 & (load4(s[31:]) >> 4)
+	s13 := 2097151 & (load3(s[34:]) >> 1)
+	s14 := 2097151 & (load4(s[36:]) >> 6)
+	s15 := 2097151 & (load3(s[39:]) >> 3)
+	s16 := 2097151 & load3(s[42:])
+	s17 := 2097151 & (load4(s[44:]) >> 5)
+	s18 := 2097151 & (load3(s[47:]) >> 2)
+	s19 := 2097151 & (load4(s[49:]) >> 7)
+	s20 := 2097151 & (load4(s[52:]) >> 4)
+	s21 := 2097151 & (load3(s[55:]) >> 1)
+	s22 := 2097151 & (load4(s[57:]) >> 6)
+	s23 := (load4(s[60:]) >> 3)
+
+	s11 += s23 * 666643
+	s12 += s23 * 470296
+	s13 += s23 * 654183
+	s14 -= s23 * 997805
+	s15 += s23 * 136657
+	s16 -= s23 * 683901
+	s23 = 0
+
+	s10 += s22 * 666643
+	s11 += s22 * 470296
+	s12 += s22 * 654183
+	s13 -= s22 * 997805
+	s14 += s22 * 136657
+	s15 -= s22 * 683901
+	s22 = 0
+
+	s9 += s21 * 666643
+	s10 += s21 * 470296
+	s11 += s21 * 654183
+	s12 -= s21 * 997805
+	s13 += s21 * 136657
+	s14 -= s21 * 683901
+	s21 = 0
+
+	s8 += s20 * 666643
+	s9 += s20 * 470296
+	s10 += s20 * 654183
+	s11 -= s20 * 997805
+	s12 += s20 * 136657
+	s13 -= s20 * 683901
+	s20 = 0
+
+	s7 += s19 * 666643
+	s8 += s19 * 470296
+	s9 += s19 * 654183
+	s10 -= s19 * 997805
+	s11 += s19 * 136657
+	s12 -= s19 * 683901
+	s19 = 0
+
+	s6 += s18 * 666643
+	s7 += s18 * 470296
+	s8 += s18 * 654183
+	s9 -= s18 * 997805
+	s10 += s18 * 136657
+	s11 -= s18 * 683901
+	s18 = 0
+
+	var carry [17]int64
+
+	carry[6] = (s6 + (1 << 20)) >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[8] = (s8 + (1 << 20)) >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[10] = (s10 + (1 << 20)) >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+	carry[12] = (s12 + (1 << 20)) >> 21
+	s13 += carry[12]
+	s12 -= carry[12] << 21
+	carry[14] = (s14 + (1 << 20)) >> 21
+	s15 += carry[14]
+	s14 -= carry[14] << 21
+	carry[16] = (s16 + (1 << 20)) >> 21
+	s17 += carry[16]
+	s16 -= carry[16] << 21
+
+	carry[7] = (s7 + (1 << 20)) >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[9] = (s9 + (1 << 20)) >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[11] = (s11 + (1 << 20)) >> 21
+	s12 += carry[11]
+	s11 -= carry[11] << 21
+	carry[13] = (s13 + (1 << 20)) >> 21
+	s14 += carry[13]
+	s13 -= carry[13] << 21
+	carry[15] = (s15 + (1 << 20)) >> 21
+	s16 += carry[15]
+	s15 -= carry[15] << 21
+
+	s5 += s17 * 666643
+	s6 += s17 * 470296
+	s7 += s17 * 654183
+	s8 -= s17 * 997805
+	s9 += s17 * 136657
+	s10 -= s17 * 683901
+	s17 = 0
+
+	s4 += s16 * 666643
+	s5 += s16 * 470296
+	s6 += s16 * 654183
+	s7 -= s16 * 997805
+	s8 += s16 * 136657
+	s9 -= s16 * 683901
+	s16 = 0
+
+	s3 += s15 * 666643
+	s4 += s15 * 470296
+	s5 += s15 * 654183
+	s6 -= s15 * 997805
+	s7 += s15 * 136657
+	s8 -= s15 * 683901
+	s15 = 0
+
+	s2 += s14 * 666643
+	s3 += s14 * 470296
+	s4 += s14 * 654183
+	s5 -= s14 * 997805
+	s6 += s14 * 136657
+	s7 -= s14 * 683901
+	s14 = 0
+
+	s1 += s13 * 666643
+	s2 += s13 * 470296
+	s3 += s13 * 654183
+	s4 -= s13 * 997805
+	s5 += s13 * 136657
+	s6 -= s13 * 683901
+	s13 = 0
+
+	s0 += s12 * 666643
+	s1 += s12 * 470296
+	s2 += s12 * 654183
+	s3 -= s12 * 997805
+	s4 += s12 * 136657
+	s5 -= s12 * 683901
+	s12 = 0
+
+	carry[0] = (s0 + (1 << 20)) >> 21
+	s1 += carry[0]
+	s0 -= carry[0] << 21
+	carry[2] = (s2 + (1 << 20)) >> 21
+	s3 += carry[2]
+	s2 -= carry[2] << 21
+	carry[4] = (s4 + (1 << 20)) >> 21
+	s5 += carry[4]
+	s4 -= carry[4] << 21
+	carry[6] = (s6 + (1 << 20)) >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[8] = (s8 + (1 << 20)) >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[10] = (s10 + (1 << 20)) >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+
+	carry[1] = (s1 + (1 << 20)) >> 21
+	s2 += carry[1]
+	s1 -= carry[1] << 21
+	carry[3] = (s3 + (1 << 20)) >> 21
+	s4 += carry[3]
+	s3 -= carry[3] << 21
+	carry[5] = (s5 + (1 << 20)) >> 21
+	s6 += carry[5]
+	s5 -= carry[5] << 21
+	carry[7] = (s7 + (1 << 20)) >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[9] = (s9 + (1 << 20)) >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[11] = (s11 + (1 << 20)) >> 21
+	s12 += carry[11]
+	s11 -= carry[11] << 21
+
+	s0 += s12 * 666643
+	s1 += s12 * 470296
+	s2 += s12 * 654183
+	s3 -= s12 * 997805
+	s4 += s12 * 136657
+	s5 -= s12 * 683901
+	s12 = 0
+
+	carry[0] = s0 >> 21
+	s1 += carry[0]
+	s0 -= carry[0] << 21
+	carry[1] = s1 >> 21
+	s2 += carry[1]
+	s1 -= carry[1] << 21
+	carry[2] = s2 >> 21
+	s3 += carry[2]
+	s2 -= carry[2] << 21
+	carry[3] = s3 >> 21
+	s4 += carry[3]
+	s3 -= carry[3] << 21
+	carry[4] = s4 >> 21
+	s5 += carry[4]
+	s4 -= carry[4] << 21
+	carry[5] = s5 >> 21
+	s6 += carry[5]
+	s5 -= carry[5] << 21
+	carry[6] = s6 >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[7] = s7 >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[8] = s8 >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[9] = s9 >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[10] = s10 >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+	carry[11] = s11 >> 21
+	s12 += carry[11]
+	s11 -= carry[11] << 21
+
+	s0 += s12 * 666643
+	s1 += s12 * 470296
+	s2 += s12 * 654183
+	s3 -= s12 * 997805
+	s4 += s12 * 136657
+	s5 -= s12 * 683901
+	s12 = 0
+
+	carry[0] = s0 >> 21
+	s1 += carry[0]
+	s0 -= carry[0] << 21
+	carry[1] = s1 >> 21
+	s2 += carry[1]
+	s1 -= carry[1] << 21
+	carry[2] = s2 >> 21
+	s3 += carry[2]
+	s2 -= carry[2] << 21
+	carry[3] = s3 >> 21
+	s4 += carry[3]
+	s3 -= carry[3] << 21
+	carry[4] = s4 >> 21
+	s5 += carry[4]
+	s4 -= carry[4] << 21
+	carry[5] = s5 >> 21
+	s6 += carry[5]
+	s5 -= carry[5] << 21
+	carry[6] = s6 >> 21
+	s7 += carry[6]
+	s6 -= carry[6] << 21
+	carry[7] = s7 >> 21
+	s8 += carry[7]
+	s7 -= carry[7] << 21
+	carry[8] = s8 >> 21
+	s9 += carry[8]
+	s8 -= carry[8] << 21
+	carry[9] = s9 >> 21
+	s10 += carry[9]
+	s9 -= carry[9] << 21
+	carry[10] = s10 >> 21
+	s11 += carry[10]
+	s10 -= carry[10] << 21
+
+	out[0] = byte(s0 >> 0)
+	out[1] = byte(s0 >> 8)
+	out[2] = byte((s0 >> 16) | (s1 << 5))
+	out[3] = byte(s1 >> 3)
+	out[4] = byte(s1 >> 11)
+	out[5] = byte((s1 >> 19) | (s2 << 2))
+	out[6] = byte(s2 >> 6)
+	out[7] = byte((s2 >> 14) | (s3 << 7))
+	out[8] = byte(s3 >> 1)
+	out[9] = byte(s3 >> 9)
+	out[10] = byte((s3 >> 17) | (s4 << 4))
+	out[11] = byte(s4 >> 4)
+	out[12] = byte(s4 >> 12)
+	out[13] = byte((s4 >> 20) | (s5 << 1))
+	out[14] = byte(s5 >> 7)
+	out[15] = byte((s5 >> 15) | (s6 << 6))
+	out[16] = byte(s6 >> 2)
+	out[17] = byte(s6 >> 10)
+	out[18] = byte((s6 >> 18) | (s7 << 3))
+	out[19] = byte(s7 >> 5)
+	out[20] = byte(s7 >> 13)
+	out[21] = byte(s8 >> 0)
+	out[22] = byte(s8 >> 8)
+	out[23] = byte((s8 >> 16) | (s9 << 5))
+	out[24] = byte(s9 >> 3)
+	out[25] = byte(s9 >> 11)
+	out[26] = byte((s9 >> 19) | (s10 << 2))
+	out[27] = byte(s10 >> 6)
+	out[28] = byte((s10 >> 14) | (s11 << 7))
+	out[29] = byte(s11 >> 1)
+	out[30] = byte(s11 >> 9)
+	out[31] = byte(s11 >> 17)
+}
+
+// GenerateKey generates a public/private key pair using randomness from rand.
+func GenerateKey(rand io.Reader) (publicKey *[PublicKeySize]byte, privateKey *[PrivateKeySize]byte, err error) {
+	privateKey = new([64]byte)
+	publicKey = new([32]byte)
+	_, err = io.ReadFull(rand, privateKey[:32])
+	if err != nil {
+		return nil, nil, err
+	}
+
+	h := sha512.New()
+	h.Write(privateKey[:32])
+	digest := h.Sum(nil)
+
+	digest[0] &= 248
+	digest[31] &= 127
+	digest[31] |= 64
+
+	var A extendedGroupElement
+	var hBytes [32]byte
+	copy(hBytes[:], digest)
+	geScalarMultBase(&A, &hBytes)
+	A.ToBytes(publicKey)
+
+	copy(privateKey[32:], publicKey[:])
+	return
+}
+
+// Sign signs the message with privateKey and returns a signature.
+func Sign(privateKey *[PrivateKeySize]byte, message []byte) *[SignatureSize]byte {
+	h := sha512.New()
+	h.Write(privateKey[:32])
+
+	var digest1, messageDigest, hramDigest [64]byte
+	var expandedSecretKey [32]byte
+	h.Sum(digest1[:0])
+	copy(expandedSecretKey[:], digest1[:])
+	expandedSecretKey[0] &= 248
+	expandedSecretKey[31] &= 63
+	expandedSecretKey[31] |= 64
+
+	h.Reset()
+	h.Write(digest1[32:])
+	h.Write(message)
+	h.Sum(messageDigest[:0])
+
+	var messageDigestReduced [32]byte
+	scReduce(&messageDigestReduced, &messageDigest)
+	var R extendedGroupElement
+	geScalarMultBase(&R, &messageDigestReduced)
+
+	var encodedR [32]byte
+	R.ToBytes(&encodedR)
+
+	h.Reset()
+	h.Write(encodedR[:])
+	h.Write(privateKey[32:])
+	h.Write(message)
+	h.Sum(hramDigest[:0])
+	var hramDigestReduced [32]byte
+	scReduce(&hramDigestReduced, &hramDigest)
+
+	var s [32]byte
+	scMulAdd(&s, &hramDigestReduced, &expandedSecretKey, &messageDigestReduced)
+
+	signature := new([64]byte)
+	copy(signature[:], encodedR[:])
+	copy(signature[32:], s[:])
+	return signature
+}
+
+// Verify returns true iff sig is a valid signature of message by publicKey.
+func Verify(publicKey *[PublicKeySize]byte, message []byte, sig *[SignatureSize]byte) bool {
+	if sig[63]&224 != 0 {
+		return false
+	}
+
+	var A extendedGroupElement
+	if !A.FromBytes(publicKey) {
+		return false
+	}
+
+	h := sha512.New()
+	h.Write(sig[:32])
+	h.Write(publicKey[:])
+	h.Write(message)
+	var digest [64]byte
+	h.Sum(digest[:0])
+
+	var hReduced [32]byte
+	scReduce(&hReduced, &digest)
+
+	var R projectiveGroupElement
+	var b [32]byte
+	copy(b[:], sig[32:])
+	geDoubleScalarMultVartime(&R, &hReduced, &A, &b)
+
+	var checkR [32]byte
+	R.ToBytes(&checkR)
+	return bytes.Equal(sig[:32], checkR[:])
+}
diff --git a/ed25519_ref.go b/ed25519_ref.go
deleted file mode 100644
index ef4dcdf..0000000
--- a/ed25519_ref.go
+++ /dev/null
@@ -1,377 +0,0 @@
-// Copyright 2012 The Go Authors. All rights reserved.
-// Use of this source code is governed by a BSD-style
-// license that can be found in the LICENSE file.
-
-// Package ed25519 implements the Ed25519 signature algorithm. See
-// http://ed25519.cr.yp.to/.
-package ed25519
-
-import (
-	"bytes"
-	"crypto/sha512"
-	"io"
-	"math/big"
-)
-
-const (
-	PublicKeySize = 32
-	PrivateKeySize = 64
-	SignatureSize = 64
-)
-
-var (
-	// p25519 is 2**255-19, a prime number.
-	p25519 *big.Int
-	// n25521 is p-2 = 2**255-21.
-	n25521 *big.Int
-	// n2523 is (p-5)/8 = 2**252-3
-	n2523 *big.Int
-	// sqrtm1 is sqrt(-1) mod p
-	sqrtm1 *big.Int
-	// order is the order of the Twisted Edward's group: 2**252 + 27742317777372353535851937790883648493.
-	order *big.Int
-	// d is -121665/121666 mod 2**255-19, a parameter of the Twisted Edward's curve.
-	d *big.Int
-	// k is 2*d mod 2**255-19
-	k *big.Int
-	// (bX, bY) is the base point of the curve.
-	bX, bY *big.Int
-	bigOne *big.Int
-)
-
-func init() {
-	p25519, _ = new(big.Int).SetString("7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffed", 16)
-	n25521, _ = new(big.Int).SetString("7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeb", 16)
-	n2523, _ = new(big.Int).SetString("ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffd", 16)
-	sqrtm1, _ = new(big.Int).SetString("2b8324804fc1df0b2b4d00993dfbd7a72f431806ad2fe478c4ee1b274a0ea0b0", 16)
-	d, _ = new(big.Int).SetString("52036cee2b6ffe738cc740797779e89800700a4d4141d8ab75eb4dca135978a3", 16)
-	k, _ = new(big.Int).SetString("2406d9dc56dffce7198e80f2eef3d13000e0149a8283b156ebd69b9426b2f159", 16)
-	bX, _ = new(big.Int).SetString("216936d3cd6e53fec0a4e231fdd6dc5c692cc7609525a7b2c9562d608f25d51a", 16)
-	bY, _ = new(big.Int).SetString("6666666666666666666666666666666666666666666666666666666666666658", 16)
-	order, _ = new(big.Int).SetString("7237005577332262213973186563042994240857116359379907606001950938285454250989", 10)
-	bigOne = big.NewInt(1)
-}
-
-// invert sets out = a^{-1} mod 2**255-19 by raising to the power of p-2 (see
-// Fermat's Little Theorem).
-func invert(out, a *big.Int) {
-	out.Exp(a, n25521, p25519)
-}
-
-// context is used to provide a pool of allocated big.Int's for use during
-// scalar multiplication.
-type context struct {
-	t1, t2, t3, t4, t5, t6, t7, t8 *big.Int
-}
-
-func newContext() *context {
-	return &context{
-		t1: new(big.Int),
-		t2: new(big.Int),
-		t3: new(big.Int),
-		t4: new(big.Int),
-		t5: new(big.Int),
-		t6: new(big.Int),
-		t7: new(big.Int),
-		t8: new(big.Int),
-	}
-}
-
-// GenerateKey generates a public/private key pair using randomness from rand.
-func GenerateKey(rand io.Reader) (publicKey *[PublicKeySize]byte, privateKey *[PrivateKeySize]byte, err error) {
-	privateKey = new([64]byte)
-	publicKey = new([32]byte)
-	_, err = io.ReadFull(rand, privateKey[:32])
-	if err != nil {
-		return nil, nil, err
-	}
-
-	h := sha512.New()
-	h.Write(privateKey[:32])
-	digest := h.Sum(nil)
-
-	digest[0] &= 248
-	digest[31] &= 127
-	digest[31] |= 64
-	a := scalarFrom32Bytes(digest[:32])
-	Ax, Ay := newContext().scalarMult(a, bX, bY).toAffine()
-	encodePoint(publicKey, Ax, Ay)
-
-	copy(privateKey[32:], publicKey[:])
-	return
-}
-
-// a fieldElement is a point on the Twisted Edward's curve represented in
-// extended coordinates. An affine point (x', y') is encoded as x = x'/z,
-// y = y'/z, t = x'y'/z.
-type fieldElement struct {
-	x, y, z, t *big.Int
-}
-
-func newFieldElement() *fieldElement {
-	return &fieldElement{
-		x: new(big.Int),
-		y: new(big.Int),
-		z: new(big.Int),
-		t: new(big.Int),
-	}
-}
-
-// toAffine converts from extended coordinates to affine coordinates.
-func (fe *fieldElement) toAffine() (x, y *big.Int) {
-	zInv := new(big.Int)
-	invert(zInv, fe.z)
-	x = new(big.Int).Mul(zInv, fe.x)
-	x.Mod(x, p25519)
-	y = new(big.Int).Mul(zInv, fe.y)
-	y.Mod(y, p25519)
-	return
-}
-
-// pointAdd adds two curve points, a and b, and puts a+b into out.
-// See http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3
-func (c *context) pointAdd(out, a, b *fieldElement) {
-	c.t1.Sub(a.y, a.x)
-	if c.t1.Sign() < 0 {
-		c.t1.Add(c.t1, p25519)
-	}
-	c.t2.Sub(b.y, b.x)
-	if c.t2.Sign() < 0 {
-		c.t2.Add(c.t2, p25519)
-	}
-	c.t1.Mul(c.t1, c.t2) // A
-
-	c.t2.Add(a.y, a.x)
-	c.t3.Add(b.y, b.x)
-	c.t2.Mul(c.t2, c.t3) // B
-
-	c.t3.Mul(a.t, b.t)
-	c.t3.Mul(c.t3, k) // C
-
-	c.t4.Mul(a.z, b.z)
-	c.t4.Lsh(c.t4, 1) // D
-
-	c.t5.Sub(c.t2, c.t1) // E
-	if c.t5.Sign() < 0 {
-		c.t5.Add(c.t5, p25519)
-	}
-
-	c.t6.Sub(c.t4, c.t3) // F
-	if c.t6.Sign() < 0 {
-		c.t6.Add(c.t6, p25519)
-	}
-
-	c.t7.Add(c.t4, c.t3) // G
-	c.t8.Add(c.t1, c.t2) // H
-
-	out.x.Mul(c.t5, c.t6)
-	out.x.Mod(out.x, p25519)
-	out.y.Mul(c.t7, c.t8)
-	out.y.Mod(out.y, p25519)
-	out.t.Mul(c.t5, c.t8)
-	out.t.Mod(out.t, p25519)
-	out.z.Mul(c.t6, c.t7)
-	out.z.Mod(out.z, p25519)
-}
-
-// scalarMult calculates scalar*(px, py) and returns the result in extended
-// coordinates.
-func (c *context) scalarMult(scalar, px, py *big.Int) *fieldElement {
-	scalarBytes := scalar.Bytes()
-	n := newFieldElement()
-	p := newFieldElement()
-	p.x.Set(px)
-	p.y.Set(py)
-	p.z.SetInt64(1)
-	p.t.Mul(p.x, p.y)
-	n.x.SetInt64(0)
-	n.y.SetInt64(1)
-	n.z.SetInt64(1)
-	n.t.SetInt64(0)
-
-	for _, x := range scalarBytes {
-		for bit := 0; bit < 8; bit++ {
-			c.pointAdd(n, n, n)
-			if x&0x80 != 0 {
-				c.pointAdd(n, n, p)
-			}
-			x <<= 1
-		}
-	}
-
-	return n
-}
-
-// encodePoint encodes a point on the curve into 32 bytes by encoding the y
-// coordinate in the first 255 bits, followed by the parity of the
-// x-coordinate.
-func encodePoint(out *[32]byte, px, py *big.Int) {
-	b := py.Bytes()
-	for i := range b {
-		out[i] = b[len(b)-(1+i)]
-	}
-	if px.Bit(0) == 1 {
-		out[31] |= 0x80
-	}
-}
-
-// scalarFrom64Bytes interprets v as a 64-byte, little-endian number and
-// returns an element of the scalar field.
-func scalarFrom64Bytes(v []byte) *big.Int {
-	// math.big takes bytes in big-endian form so we have to reverse the
-	// bytes.
-	var reversedBytes [64]byte
-	for i := 0; i < 64; i++ {
-		reversedBytes[i] = v[63-i]
-	}
-	r := new(big.Int).SetBytes(reversedBytes[:])
-	r.Mod(r, order)
-	return r
-}
-
-// scalarFrom32Bytes interprets v as a 32-byte, little-endian number and
-// returns an element of the scalar field.
-func scalarFrom32Bytes(v []byte) *big.Int {
-	// math.big takes bytes in big-endian form so we have to reverse the
-	// bytes.
-	var reversedBytes [32]byte
-	for i := 0; i < 32; i++ {
-		reversedBytes[i] = v[31-i]
-	}
-	n := new(big.Int).SetBytes(reversedBytes[:])
-	n.Mod(n, order)
-	return n
-}
-
-// encodeScalar encodes s in to out as a 32-byte, little-endian number.
-func encodeScalar(out []byte, s *big.Int) {
-	b := s.Bytes()
-	for i := 0; i < len(b); i++ {
-		out[i] = b[len(b)-(1+i)]
-	}
-}
-
-// Sign signs the message with privateKey and returns a signature.
-func Sign(privateKey *[PrivateKeySize]byte, message []byte) *[SignatureSize]byte {
-	h := sha512.New()
-	h.Write(privateKey[:32])
-	var digestBytes1, digestBytes2, digestBytes3 [64]byte
-	expandedSecretKey := h.Sum(digestBytes1[:0])
-
-	expandedSecretKey[0] &= 248
-	expandedSecretKey[31] &= 127
-	expandedSecretKey[31] |= 64
-	a := scalarFrom32Bytes(expandedSecretKey[:32])
-
-	h.Reset()
-	h.Write(expandedSecretKey[32:])
-	h.Write(message)
-	messageDigest := h.Sum(digestBytes2[:0])
-
-	r := scalarFrom64Bytes(messageDigest)
-	Rx, Ry := newContext().scalarMult(r, bX, bY).toAffine()
-	var encodedR [32]byte
-	encodePoint(&encodedR, Rx, Ry)
-
-	h.Reset()
-	h.Write(encodedR[:])
-	h.Write(privateKey[32:])
-	h.Write(message)
-	hramDigest := h.Sum(digestBytes3[:0])
-	s := scalarFrom64Bytes(hramDigest)
-	s.Mul(s, a)
-	s.Add(s, r)
-	s.Mod(s, order)
-
-	signature := new([64]byte)
-	copy(signature[:32], encodedR[:])
-	encodeScalar(signature[32:], s)
-	return signature
-}
-
-// decodePoint unpacks a curve point, as encoded by encodePoint, and returns
-// its affine coordinates.
-func decodePoint(in *[32]byte) (x, y *big.Int, ok bool) {
-	var bigEndian [32]byte
-	for i := 0; i < 32; i++ {
-		bigEndian[i] = in[31-i]
-	}
-	bigEndian[0] &= 0x7f
-	y = new(big.Int).SetBytes(bigEndian[:])
-
-	u := new(big.Int).Mul(y, y)
-	u.Mod(u, p25519)
-	v := new(big.Int).Set(u)
-	u.Sub(u, bigOne)
-	if u.Sign() < 0 {
-		u.Add(u, p25519)
-	}
-	v.Mul(v, d)
-	v.Add(v, bigOne)
-	v.Mod(v, p25519)
-
-	v2 := new(big.Int).Mul(v, v)
-	v3 := new(big.Int).Mul(v2, v)
-	v3.Mod(v3, p25519)
-	v4 := new(big.Int).Mul(v2, v2)
-	v7 := v4.Mul(v4, v3)
-	b := v7.Mul(v7, u)
-	b.Mod(b, p25519)
-	b.Exp(b, n2523, p25519)
-	b.Mul(b, v3)
-	b.Mul(b, u)
-	b.Mod(b, p25519)
-
-	check := v2.Mul(b, b)
-	check.Mul(check, v)
-	check.Mod(check, p25519)
-	if check.Cmp(u) != 0 {
-		b.Mul(b, sqrtm1)
-		b.Mod(b, p25519)
-		check.Mul(b, b)
-		check.Mul(check, v)
-		check.Mod(check, p25519)
-	}
-
-	if check.Cmp(u) != 0 {
-		return nil, nil, false
-	}
-
-	if b.Bit(0) != uint(in[31]>>7) {
-		b.Neg(b)
-		b.Add(b, p25519)
-	}
-
-	return b, y, true
-}
-
-// Verify returns true iff sig is a valid signature of message by publicKey.
-func Verify(publicKey *[PublicKeySize]byte, message []byte, sig *[SignatureSize]byte) bool {
-	h := sha512.New()
-	h.Write(sig[:32])
-	h.Write(publicKey[:])
-	h.Write(message)
-	digest := h.Sum(nil)
-	H := scalarFrom64Bytes(digest)
-	S := scalarFrom32Bytes(sig[32:])
-
-	Ax, Ay, ok := decodePoint(publicKey)
-	if !ok {
-		return false
-	}
-
-	c := newContext()
-	SB := c.scalarMult(S, bX, bY)
-	HRAMA := c.scalarMult(H, Ax, Ay)
-	HRAMA.x.Neg(HRAMA.x)
-	HRAMA.x.Add(HRAMA.x, p25519)
-	HRAMA.t.Neg(HRAMA.t)
-	HRAMA.t.Add(HRAMA.t, p25519)
-	out := newFieldElement()
-	c.pointAdd(out, SB, HRAMA)
-	Px, Py := out.toAffine()
-
-	var rPrime [32]byte
-	encodePoint(&rPrime, Px, Py)
-	return bytes.Equal(sig[:32], rPrime[:])
-}
diff --git a/ed25519_test.go b/ed25519_test.go
index b9580f5..0dc2b13 100644
--- a/ed25519_test.go
+++ b/ed25519_test.go
@@ -25,7 +25,6 @@ func (zeroReader) Read(buf []byte) (int, error) {
 }
 
 func TestSignVerify(t *testing.T) {
-	return
 	var zero zeroReader
 	public, private, _ := GenerateKey(zero)
 
@@ -94,7 +93,7 @@ func TestGolden(t *testing.T) {
 
 		sig2 := Sign(&priv, msg)
 		if !bytes.Equal(sig, sig2[:]) {
-			t.Errorf("different signature result on line %d", lineNo)
+			t.Errorf("different signature result on line %d: %x vs %x", lineNo, sig, sig2)
 		}
 
 		var pubKey [PublicKeySize]byte

-- 
Alioth's /usr/local/bin/git-commit-notice on /srv/git.debian.org/git/pkg-privacy/packages/golang-ed25519-dev.git



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